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Elements of the Differential and Integral Calculus - Granville - Revised.djvu

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Cover frontispiece Title  iv  v  vi  vii  viii  ix  x  xi  xii  xiii  xiv  xv Image 001 002 003 004 005 006 007 008 009 010 011 012 013 014 015 016 017 018 019 020 021 022 023 024 025 026 027 028 029 030 031 032 033 034 035 036 037 038 039 040 041 042 043 044 045 046 047 048 049 050 051 052 053 054 055 056 057 058 059 060 061 062 063 064 065 066 067 068 069 070 071 072 073 074 075 076 077 078 079 080 081 082 083 084 085 086 087 088 089 090 091 092 093 094 095 096 097 098 099 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468

  1. CONTENTS
     

    DIFFERENTIAL CALCULUS

     

    CHAPTER I
    COLLECTION OF FORMULAS

  2. SECTION PAGE
  3. 1.
    Formulas from Algebra, Trigonometry, and Analytic Geometry
    ................................................................................................................................................................................................................................................................................................................................................................................................
    1
  4. 2.
    Greek alphabet
    ................................................................................................................................................................................................................................................................................................................................................................................................
    3
  5. 3.
    Rules for signs in the four quadrants
    ................................................................................................................................................................................................................................................................................................................................................................................................
    3
  6. 4.
    Natural values of the trigonometric functions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    4
  7. 5.
    Tables of logarithms
    ................................................................................................................................................................................................................................................................................................................................................................................................
    5
  8.  

    CHAPTER II
    VARIABLES AND FUNCTIONS

  9. 6.
    Variables and constants
    ................................................................................................................................................................................................................................................................................................................................................................................................
    6
  10. 7.
    Interval of a variable
    ................................................................................................................................................................................................................................................................................................................................................................................................
    6
  11. 8.
    Continuous variation
    ................................................................................................................................................................................................................................................................................................................................................................................................
    6
  12. 9.
    Functions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    7
  13. 10.
    Independent and dependent variables
    ................................................................................................................................................................................................................................................................................................................................................................................................
    7
  14. 11.
    Notation of functions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    8
  15. 12.
    Values of the independent variable for which a function is defined
    ................................................................................................................................................................................................................................................................................................................................................................................................
    8
  16.  

    CHAPTER III
    THEORY OF LIMITS

  17. 13.
    Limit of a variable
    ................................................................................................................................................................................................................................................................................................................................................................................................
    11
  18. 14.
    Division by zero excluded
    ................................................................................................................................................................................................................................................................................................................................................................................................
    12
  19. 15.
    Infinitesimals
    ................................................................................................................................................................................................................................................................................................................................................................................................
    13
  20. 16.
    The concept of infinity (\scriptstyle{\infty})
    ................................................................................................................................................................................................................................................................................................................................................................................................
    13
  21. 17.
    Limiting value of a function
    ................................................................................................................................................................................................................................................................................................................................................................................................
    14
  22. 18.
    Continuous and discontinuous functions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    14
  23. 19.
    Continuity and discontinuity of functions illustrated by their graphs
    ................................................................................................................................................................................................................................................................................................................................................................................................
    16
  24. 20.
    Fundamental theorems on limits
    ................................................................................................................................................................................................................................................................................................................................................................................................
    18
  25. 21.
    Special limiting values
    ................................................................................................................................................................................................................................................................................................................................................................................................
    20
  26. 22.
    The limit of \scriptstyle{\frac{\sin x}{x}} as \scriptstyle{x\doteq0}
    ................................................................................................................................................................................................................................................................................................................................................................................................
    21
  27. 23.
    The number \scriptstyle{e}
    ................................................................................................................................................................................................................................................................................................................................................................................................
    22
  28. 24.
    Expressions assuming the form \scriptstyle{\frac{\infty}{\infty}}
    ................................................................................................................................................................................................................................................................................................................................................................................................
    23
  29.  
  30. CHAPTER IV

    DIFFERENTIATION

  31. 25.
    Introduction
    ................................................................................................................................................................................................................................................................................................................................................................................................
    25
  32. 26.
    Increments
    ................................................................................................................................................................................................................................................................................................................................................................................................
    25
  33. 27.
    Comparison of increments
    ................................................................................................................................................................................................................................................................................................................................................................................................
    26
  34. 28.
    Derivative of a function of one variable
    ................................................................................................................................................................................................................................................................................................................................................................................................
    27
  35. 29.
    Symbols for derivatives
    ................................................................................................................................................................................................................................................................................................................................................................................................
    28
  36. 30.
    Differentiable functions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    29
  37. 31.
    General rule for differentiation
    ................................................................................................................................................................................................................................................................................................................................................................................................
    29
  38. 32.
    Applications of the derivative to Geometry
    ................................................................................................................................................................................................................................................................................................................................................................................................
    31
  39.  

    CHAPTER V
    RULES FOR DIFFERENTIATING STANDARD ELEMENTARY FORMS

  40. 33.
    Importance of General Rule
    ................................................................................................................................................................................................................................................................................................................................................................................................
    34
  41. 34.
    Differentiation of a constant
    ................................................................................................................................................................................................................................................................................................................................................................................................
    36
  42. 35.
    Differentiation of a variable with respect to itself
    ................................................................................................................................................................................................................................................................................................................................................................................................
    37
  43. 36.
    Differentiation of a sum
    ................................................................................................................................................................................................................................................................................................................................................................................................
    37
  44. 37.
    Differentiation of the product of a constant and a function
    ................................................................................................................................................................................................................................................................................................................................................................................................
    37
  45. 38.
    Differentiation of the product of two functions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    38
  46. 39.
    Differentiation of the product of any finite number of functions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    38
  47. 40.
    Differentiation of a function with a constant exponent
    ................................................................................................................................................................................................................................................................................................................................................................................................
    39
  48. 41.
    Differentiation of a quotient
    ................................................................................................................................................................................................................................................................................................................................................................................................
    40
  49. 42.
    Differentiation of a function of a function
    ................................................................................................................................................................................................................................................................................................................................................................................................
    44
  50. 43.
    Differentiation of inverse functions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    45
  51. 44.
    Differentiation of a logarithm
    ................................................................................................................................................................................................................................................................................................................................................................................................
    46
  52. 45.
    Differentiation of the simple exponential function
    ................................................................................................................................................................................................................................................................................................................................................................................................
    48
  53. 46.
    Differentiation of the general exponential function
    ................................................................................................................................................................................................................................................................................................................................................................................................
    49
  54. 47.
    Logarithmic differentiation
    ................................................................................................................................................................................................................................................................................................................................................................................................
    50
  55. 48.
    Differentiation of \scriptstyle{\sin v}
    ................................................................................................................................................................................................................................................................................................................................................................................................
    54
  56. 49.
    Differentiation of \scriptstyle{\cos v}
    ................................................................................................................................................................................................................................................................................................................................................................................................
    55
  57. 50.
    Differentiation of \scriptstyle{\tan v}
    ................................................................................................................................................................................................................................................................................................................................................................................................
    56
  58. 51.
    Differentiation of \scriptstyle{\cot v}
    ................................................................................................................................................................................................................................................................................................................................................................................................
    56
  59. 52.
    Differentiation of \scriptstyle{\sec v}
    ................................................................................................................................................................................................................................................................................................................................................................................................
    56
  60. 53.
    Differentiation of \scriptstyle{\csc v}
    ................................................................................................................................................................................................................................................................................................................................................................................................
    57
  61. 54.
    Differentiation of \scriptstyle{\operatorname{vers} v}
    ................................................................................................................................................................................................................................................................................................................................................................................................
    57
  62. 55.
    Differentiation of \scriptstyle{\operatorname{arc~sin} v}
    ................................................................................................................................................................................................................................................................................................................................................................................................
    61
  63. 56.
    Differentiation of \scriptstyle{\operatorname{arc~cos} v}
    ................................................................................................................................................................................................................................................................................................................................................................................................
    62
  64. 57.
    Differentiation of \scriptstyle{\operatorname{arc~tan} v}
    ................................................................................................................................................................................................................................................................................................................................................................................................
    62
  65. 58.
    Differentiation of \scriptstyle{\operatorname{arc~cot} v}
    ................................................................................................................................................................................................................................................................................................................................................................................................
    63
  66. 59.
    Differentiation of \scriptstyle{\operatorname{arc~sec} v}
    ................................................................................................................................................................................................................................................................................................................................................................................................
    63
  67. 60.
    Differentiation of \scriptstyle{\operatorname{arc~csc} v}
    ................................................................................................................................................................................................................................................................................................................................................................................................
    64
  68. 61.
    Differentiation of \scriptstyle{\operatorname{arc~vers} v}
    ................................................................................................................................................................................................................................................................................................................................................................................................
    65
  69. 62.
    Implicit functions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    69
  70. 63.
    Differentiation of implicit functions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    69
  71.  

    CHAPTER VI
    SIMPLE APPLICATIONS OF THE DERIVATIVE

  72. 64.
    Direction of a curve
    ................................................................................................................................................................................................................................................................................................................................................................................................
    73
  73. 65.
    Equations of tangent and normal, lengths of subtangent and subnormal. Rectangular coördinates
    ................................................................................................................................................................................................................................................................................................................................................................................................
    76
  74. 66.
    Parametric equations of a curve
    ................................................................................................................................................................................................................................................................................................................................................................................................
    79
  75. 67.
    Angle between the radius vector drawn to a point on a curve and the tangent to the curve at that point
    ................................................................................................................................................................................................................................................................................................................................................................................................
    83
  76. 68.
    Lengths of polar subtangent and polar subnormal
    ................................................................................................................................................................................................................................................................................................................................................................................................
    86
  77. 69.
    Solution of equations having multiple roots
    ................................................................................................................................................................................................................................................................................................................................................................................................
    88
  78. 70.
    Applications of the derivative in mechanics. Velocity
    ................................................................................................................................................................................................................................................................................................................................................................................................
    90
  79. 71.
    Component velocities
    ................................................................................................................................................................................................................................................................................................................................................................................................
    91
  80. 72.
    Acceleration
    ................................................................................................................................................................................................................................................................................................................................................................................................
    92
  81. 73.
    Component accelerations
    ................................................................................................................................................................................................................................................................................................................................................................................................
    93
  82.  

    CHAPTER VII
    SUCCESSIVE DIFFERENTIATION

  83. 74.
    Definition of successive derivatives
    ................................................................................................................................................................................................................................................................................................................................................................................................
    97
  84. 75.
    Notation
    ................................................................................................................................................................................................................................................................................................................................................................................................
    97
  85. 76.
    The \scriptstyle{n\text{th}} derivative
    ................................................................................................................................................................................................................................................................................................................................................................................................
    98
  86. 77.
    Leibnitz's formula for the \scriptstyle{n\text{th}} derivative of a product
    ................................................................................................................................................................................................................................................................................................................................................................................................
    98
  87. 78.
    Successive differentiation of implicit functions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    100
  88.  

    CHAPTER VIII
    MAXIMA AND MINIMA. POINTS OF INFLECTION. CURVE TRACING

  89. 79.
    Introduction
    ................................................................................................................................................................................................................................................................................................................................................................................................
    103
  90. 80.
    Increasing and decreasing functions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    106
  91. 81.
    Tests for determining when a function is increasing and when decreasing
    ................................................................................................................................................................................................................................................................................................................................................................................................
    108
  92. 82.
    Maximum and minimum values of a function
    ................................................................................................................................................................................................................................................................................................................................................................................................
    109
  93. 83.
    First method for examining a function for maximum and minimum values
    ................................................................................................................................................................................................................................................................................................................................................................................................
    111
  94. 84.
    Second method for examining a function for maximum and minimum values
    ................................................................................................................................................................................................................................................................................................................................................................................................
    112
  95. 85.
    Definition of points of inflection and rule for finding points of inflection
    ................................................................................................................................................................................................................................................................................................................................................................................................
    125
  96. 86.
    Curve tracing
    ................................................................................................................................................................................................................................................................................................................................................................................................
    128
  97.  

    CHAPTER IX
    DIFFERENTIALS

  98. 87.
    Introduction
    ................................................................................................................................................................................................................................................................................................................................................................................................
    131
  99. 88.
    Definitions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    131
  100. 89.
    Infinitesimals
    ................................................................................................................................................................................................................................................................................................................................................................................................
    132
  101. 90.
    Derivative of the arc in rectangular coördinates
    ................................................................................................................................................................................................................................................................................................................................................................................................
    134
  102. 91.
    Derivative of the arc in polar coördinates
    ................................................................................................................................................................................................................................................................................................................................................................................................
    135
  103. 92.
    Formulas for finding the differentials of functions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    137
  104. 93.
    Successive differentials
    ................................................................................................................................................................................................................................................................................................................................................................................................
    139
  105.  

    CHAPTER X
    RATES

  106. 94.
    The derivative considered as the ratio of two rates
    ................................................................................................................................................................................................................................................................................................................................................................................................
    141
  107.  

    CHAPTER XI
    CHANGE OF VARIABLE

  108. 95.
    Interchange of dependent and independent variables
    ................................................................................................................................................................................................................................................................................................................................................................................................
    148
  109. 96.
    Change of the dependent variable
    ................................................................................................................................................................................................................................................................................................................................................................................................
    149
  110. 97.
    Change of the independent variable
    ................................................................................................................................................................................................................................................................................................................................................................................................
    150
  111. 98.
    Simultaneous change of both independent and dependent variables
    ................................................................................................................................................................................................................................................................................................................................................................................................
    152
  112.  

    CHAPTER XII
    CURVATURE. RADIUS OF CURVATURE

  113. 99.
    Curvature
    ................................................................................................................................................................................................................................................................................................................................................................................................
    155
  114. 100.
    Curvature of a circle
    ................................................................................................................................................................................................................................................................................................................................................................................................
    155
  115. 101.
    Curvature at a point
    ................................................................................................................................................................................................................................................................................................................................................................................................
    156
  116. 102.
    Formulas for curvature
    ................................................................................................................................................................................................................................................................................................................................................................................................
    159
  117. 103.
    Radius of curvature
    ................................................................................................................................................................................................................................................................................................................................................................................................
    159
  118. 104.
    Circle of curvature
    ................................................................................................................................................................................................................................................................................................................................................................................................
    161
  119.  

    CHAPTER XIII
    THEOREM OF MEAN VALUE. INDETERMINATE FORMS

  120. 105.
    Rolle's Theorem
    ................................................................................................................................................................................................................................................................................................................................................................................................
    164
  121. 106.
    The Theorem of Mean Value
    ................................................................................................................................................................................................................................................................................................................................................................................................
    165
  122. 107.
    The Extended Theorem of Mean Value
    ................................................................................................................................................................................................................................................................................................................................................................................................
    166
  123. 108.
    Maxima and minima treated analytically
    ................................................................................................................................................................................................................................................................................................................................................................................................
    167
  124. 109.
    Indeterminate forms
    ................................................................................................................................................................................................................................................................................................................................................................................................
    170
  125. 110.
    Evaluation of a function taking on an indeterminate form
    ................................................................................................................................................................................................................................................................................................................................................................................................
    170
  126. 111.
    Evaluation of the indeterminate form \scriptstyle{\frac{0}{0}}
    ................................................................................................................................................................................................................................................................................................................................................................................................
    171
  127. 112.
    Evaluation of the indeterminate form \scriptstyle{\frac{\infty}{\infty}}
    ................................................................................................................................................................................................................................................................................................................................................................................................
    174
  128. 113.
    Evaluation of the indeterminate form \scriptstyle{0\cdot\infty}
    ................................................................................................................................................................................................................................................................................................................................................................................................
    174
  129. 114.
    Evaluation of the indeterminate form \scriptstyle{\infty-\infty}
    ................................................................................................................................................................................................................................................................................................................................................................................................
    175
  130. 115.
    Evaluation of the indeterminate forms \scriptstyle{0^0,~1^\infty,~\infty^0}
    ................................................................................................................................................................................................................................................................................................................................................................................................
    176
  131.  

    CHAPTER XIV
    CIRCLE OF CURVATURE. CENTER OF CURVATURE

  132. 116.
    Circle of curvature. Center of curvature
    ................................................................................................................................................................................................................................................................................................................................................................................................
    178
  133. 117.
    Second method for finding center of curvature
    ................................................................................................................................................................................................................................................................................................................................................................................................
    180
  134. 118.
    Center of curvature the limiting position of the intersection of normals at neighboring points
    ................................................................................................................................................................................................................................................................................................................................................................................................
    181
  135. 119.
    Evolutes
    ................................................................................................................................................................................................................................................................................................................................................................................................
    182
  136. 120.
    Properties of the evolute
    ................................................................................................................................................................................................................................................................................................................................................................................................
    186
  137. 121.
    Involutes and their mechanical construction
    ................................................................................................................................................................................................................................................................................................................................................................................................
    187
  138.  

    CHAPTER XV
    PARTIAL DIFFERENTIATION

  139. 122.
    Continuous functions of two or more independent variables
    ................................................................................................................................................................................................................................................................................................................................................................................................
    190
  140. 123.
    Partial derivatives
    ................................................................................................................................................................................................................................................................................................................................................................................................
    191
  141. 124.
    Partial derivatives interpreted geometrically
    ................................................................................................................................................................................................................................................................................................................................................................................................
    192
  142. 125.
    Total derivatives
    ................................................................................................................................................................................................................................................................................................................................................................................................
    194
  143. 126.
    Total differentials
    ................................................................................................................................................................................................................................................................................................................................................................................................
    197
  144. 127.
    Differentiation of implicit functions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    198
  145. 128.
    Successive partial derivatives
    ................................................................................................................................................................................................................................................................................................................................................................................................
    202
  146. 129.
    Order of differentiation immaterial
    ................................................................................................................................................................................................................................................................................................................................................................................................
    203
  147.  

    CHAPTER XVI
    ENVELOPES

  148. 130.
    Family of curves. Variable parameter
    ................................................................................................................................................................................................................................................................................................................................................................................................
    205
  149. 131.
    Envelope of a family of curves depending on one parameter
    ................................................................................................................................................................................................................................................................................................................................................................................................
    205
  150. 132.
    The evolute of a given curve considered as the envelope of its normals
    ................................................................................................................................................................................................................................................................................................................................................................................................
    208
  151. 133.
    Two parameters connected by one equation of condition
    ................................................................................................................................................................................................................................................................................................................................................................................................
    209
  152.  

    CHAPTER XVII
    SERIES

  153. 134.
    Introduction
    ................................................................................................................................................................................................................................................................................................................................................................................................
    212
  154. 135.
    Infinite series
    ................................................................................................................................................................................................................................................................................................................................................................................................
    213
  155. 136.
    Existence of a limit
    ................................................................................................................................................................................................................................................................................................................................................................................................
    215
  156. 137.
    Fundamental test for convergence
    ................................................................................................................................................................................................................................................................................................................................................................................................
    216
  157. 138.
    Comparison test for convergence
    ................................................................................................................................................................................................................................................................................................................................................................................................
    217
  158. 139.
    Cauchy's ratio test for convergence
    ................................................................................................................................................................................................................................................................................................................................................................................................
    218
  159. 140.
    Alternating series
    ................................................................................................................................................................................................................................................................................................................................................................................................
    220
  160. 141.
    Absolute convergence
    ................................................................................................................................................................................................................................................................................................................................................................................................
    220
  161. 142.
    Power series
    ................................................................................................................................................................................................................................................................................................................................................................................................
    223
  162.  

    CHAPTER XVIII
    EXPANSION OF FUNCTIONS

  163. 143.
    Introduction
    ................................................................................................................................................................................................................................................................................................................................................................................................
    227
  164. 144.
    Taylor's Theorem and Taylor's Series
    ................................................................................................................................................................................................................................................................................................................................................................................................
    228
  165. 145.
    Maclaurin's Theorem and Maclaurin's Series
    ................................................................................................................................................................................................................................................................................................................................................................................................
    230
  166. 146.
    Computation by series
    ................................................................................................................................................................................................................................................................................................................................................................................................
    234
  167. 147.
    Approximate formulas derived from series. Interpolation
    ................................................................................................................................................................................................................................................................................................................................................................................................
    237
  168. 148.
    Taylor's Theorem for functions of two or more variables
    ................................................................................................................................................................................................................................................................................................................................................................................................
    240
  169. 149.
    Maxima and minima of functions of two independent variables
    ................................................................................................................................................................................................................................................................................................................................................................................................
    243
  170.  

    CHAPTER XIX
    ASYMPTOTES. SINGULAR POINTS

  171. 150.
    Rectilinear asymptotes
    ................................................................................................................................................................................................................................................................................................................................................................................................
    249
  172. 151.
    Asymptotes found by method of limiting intercepts
    ................................................................................................................................................................................................................................................................................................................................................................................................
    249
  173. 152.
    Method of determining asymptotes to algebraic curves
    ................................................................................................................................................................................................................................................................................................................................................................................................
    250
  174. 153.
    Asymptotes in polar coördinates
    ................................................................................................................................................................................................................................................................................................................................................................................................
    254
  175. 154.
    Singular points
    ................................................................................................................................................................................................................................................................................................................................................................................................
    255
  176. 155.
    Determination of the tangent to an algebraic curve at a given point by inspection
    ................................................................................................................................................................................................................................................................................................................................................................................................
    255
  177. 156.
    Nodes
    ................................................................................................................................................................................................................................................................................................................................................................................................
    258
  178. 157.
    Cusps
    ................................................................................................................................................................................................................................................................................................................................................................................................
    259
  179. 158.
    Conjugate or isolated points
    ................................................................................................................................................................................................................................................................................................................................................................................................
    260
  180. 159.
    Transcendental singularities
    ................................................................................................................................................................................................................................................................................................................................................................................................
    260
  181.  

    CHAPTER XX
    APPLICATIONS TO GEOMETRY OF SPACE

  182. 160.
    Tangent line and normal plane to a skew curve whose equations are given in parametric form
    ................................................................................................................................................................................................................................................................................................................................................................................................
    262
  183. 161.
    Tangent plane to a surface
    ................................................................................................................................................................................................................................................................................................................................................................................................
    264
  184. 162.
    Normal line to a surface
    ................................................................................................................................................................................................................................................................................................................................................................................................
    266
  185. 163.
    Another form of the equations of the tangent line to a skew curve
    ................................................................................................................................................................................................................................................................................................................................................................................................
    268
  186. 164.
    Another form of the equation of the normal plane to a skew curve
    ................................................................................................................................................................................................................................................................................................................................................................................................
    269
  187.  

    CHAPTER XXI
    CURVES FOR REFERENCE

     

     

    INTEGRAL CALCULUS

     

    CHAPTER XXII
    INTEGRATION. RULES FOR INTEGRATING STANDARD ELEMENTARY FORMS

  188. 165.
    Integration
    ................................................................................................................................................................................................................................................................................................................................................................................................
    279
  189. 166.
    Constant of integration. Indefinite integral
    ................................................................................................................................................................................................................................................................................................................................................................................................
    281
  190. 167.
    Rules for integrating standard elementary forms
    ................................................................................................................................................................................................................................................................................................................................................................................................
    282
  191. 168.
    Trigonometric differentials
    ................................................................................................................................................................................................................................................................................................................................................................................................
    298
  192. 169.
    Integration of expressions containing \scriptstyle{\sqrt{a^2-x^2}} or \scriptstyle{\sqrt{x^2\pm a^2}} by a trigonometric substitution
    ................................................................................................................................................................................................................................................................................................................................................................................................
    304
  193.  

    CHAPTER XXIII
    CONSTANT OF INTEGRATION

  194. 170.
    Determination of the constant of integration by means of initial conditions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    307
  195. 171.
    Geometrical signification of the constant of integration
    ................................................................................................................................................................................................................................................................................................................................................................................................
    307
  196. 172.
    Physical signification of the constant of integration
    ................................................................................................................................................................................................................................................................................................................................................................................................
    309
  197.  

    CHAPTER XXIV
    THE DEFINITE INTEGRAL

  198. 173.
    Differential of an area
    ................................................................................................................................................................................................................................................................................................................................................................................................
    314
  199. 174.
    The definite integral
    ................................................................................................................................................................................................................................................................................................................................................................................................
    314
  200. 175.
    Calculation of a definite integral
    ................................................................................................................................................................................................................................................................................................................................................................................................
    316
  201. 176.
    Calculation of areas
    ................................................................................................................................................................................................................................................................................................................................................................................................
    318
  202. 177.
    Geometrical representation of an integral
    ................................................................................................................................................................................................................................................................................................................................................................................................
    319
  203. 178.
    Mean value of \scriptstyle{\phi(x)}
    ................................................................................................................................................................................................................................................................................................................................................................................................
    320
  204. 179.
    Interchange of limits
    ................................................................................................................................................................................................................................................................................................................................................................................................
    320
  205. 180.
    Decomposition of the interval
    ................................................................................................................................................................................................................................................................................................................................................................................................
    321
  206. 181.
    The definite integral a function of its limits
    ................................................................................................................................................................................................................................................................................................................................................................................................
    321
  207. 182.
    Infinite limits
    ................................................................................................................................................................................................................................................................................................................................................................................................
    321
  208. 183.
    When \scriptstyle{y=\phi(x)} is discontinuous
    ................................................................................................................................................................................................................................................................................................................................................................................................
    322
  209.  

    CHAPTER XXV
    INTEGRATION OF RATIONAL FRACTIONS

  210. 184.
    Introduction
    ................................................................................................................................................................................................................................................................................................................................................................................................
    325
  211. 185.
    Case I
    ................................................................................................................................................................................................................................................................................................................................................................................................
    325
  212. 186.
    Case II
    ................................................................................................................................................................................................................................................................................................................................................................................................
    327
  213. 187.
    Case III
    ................................................................................................................................................................................................................................................................................................................................................................................................
    329
  214. 188.
    Case IV
    ................................................................................................................................................................................................................................................................................................................................................................................................
    331
  215.  

    CHAPTER XXVI
    INTEGRATION BY SUBSTITUTION OF A NEW VARIABLE. RATIONALIZATION

  216. 189.
    Introduction
    ................................................................................................................................................................................................................................................................................................................................................................................................
    335
  217. 190.
    Differentials containing fractional powers of \scriptstyle{x} only
    ................................................................................................................................................................................................................................................................................................................................................................................................
    335
  218. 191.
    Differentials containing fractional powers of \scriptstyle{a+bx} only
    ................................................................................................................................................................................................................................................................................................................................................................................................
    336
  219. 192.
    Change in limits corresponding to change in variable
    ................................................................................................................................................................................................................................................................................................................................................................................................
    336
  220. 193.
    Differentials containing no radical except \scriptstyle{\sqrt{a+bx+x^2}}
    ................................................................................................................................................................................................................................................................................................................................................................................................
    338
  221. 194.
    Differentials containing no radical except \scriptstyle{\sqrt{a+bx-x^2}}
    ................................................................................................................................................................................................................................................................................................................................................................................................
    338
  222. 195.
    Binomial differentials
    ................................................................................................................................................................................................................................................................................................................................................................................................
    340
  223. 196.
    Conditions of integrability of binomial differentials
    ................................................................................................................................................................................................................................................................................................................................................................................................
    341
  224. 197.
    Transformation of trigonometric differentials
    ................................................................................................................................................................................................................................................................................................................................................................................................
    343
  225. 198.
    Miscellaneous substitutions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    345
  226.  

    CHAPTER XXVII
    INTEGRATION BY PARTS. REDUCTION FORMULAS

  227. 199.
    Formula for integration by parts
    ................................................................................................................................................................................................................................................................................................................................................................................................
    347
  228. 200.
    Reduction formulas for binomial differentials
    ................................................................................................................................................................................................................................................................................................................................................................................................
    350
  229. 201.
    Reduction formulas for trigonometric differentials
    ................................................................................................................................................................................................................................................................................................................................................................................................
    356
  230. 202.
    To find \scriptstyle{\int e^{ax}\sin{nx}dx} and \scriptstyle{\int e^{ax}\cos{nx}dx}
    ................................................................................................................................................................................................................................................................................................................................................................................................
    359
  231.  

    CHAPTER XXVIII
    INTEGRATION A PROCESS OF SUMMATION

  232. 203.
    Introduction
    ................................................................................................................................................................................................................................................................................................................................................................................................
    361
  233. 204.
    The fundamental theorem of Integral Calculus
    ................................................................................................................................................................................................................................................................................................................................................................................................
    361
  234. 205.
    Analytical proof of the Fundamental Theorem
    ................................................................................................................................................................................................................................................................................................................................................................................................
    364
  235. 206.
    Areas of plane curves. Rectangular coördinates
    ................................................................................................................................................................................................................................................................................................................................................................................................
    365
  236. 207.
    Area when curve is given in parametric form
    ................................................................................................................................................................................................................................................................................................................................................................................................
    368
  237. 208.
    Areas of plane curves. Polar coördinates
    ................................................................................................................................................................................................................................................................................................................................................................................................
    370
  238. 209.
    Length of a curve
    ................................................................................................................................................................................................................................................................................................................................................................................................
    372
  239. 210.
    Lengths of plane curves. Rectangular coördinates
    ................................................................................................................................................................................................................................................................................................................................................................................................
    373
  240. 211.
    Lengths of plane curves. Polar coördinates
    ................................................................................................................................................................................................................................................................................................................................................................................................
    375
  241. 212.
    Volumes of solids of revolution
    ................................................................................................................................................................................................................................................................................................................................................................................................
    377
  242. 213.
    Areas of surfaces of revolution
    ................................................................................................................................................................................................................................................................................................................................................................................................
    381
  243. 214.
    Miscellaneous applications
    ................................................................................................................................................................................................................................................................................................................................................................................................
    385
  244.  

    CHAPTER XXIX
    SUCCESSIVE AND PARTIAL INTEGRATION

  245. 215.
    Successive integration
    ................................................................................................................................................................................................................................................................................................................................................................................................
    393
  246. 216.
    Partial integration
    ................................................................................................................................................................................................................................................................................................................................................................................................
    395
  247. 217.
    Definite double integral. Geometric interpretation
    ................................................................................................................................................................................................................................................................................................................................................................................................
    396
  248. 218.
    Value of a definite double integral over a region
    ................................................................................................................................................................................................................................................................................................................................................................................................
    400
  249. 219.
    Plane area as a definite double integral. Rectangular coördinates
    ................................................................................................................................................................................................................................................................................................................................................................................................
    402
  250. 220.
    Plane area as a definite double integral. Polar coördinates
    ................................................................................................................................................................................................................................................................................................................................................................................................
    406
  251. 221.
    Moment of area
    ................................................................................................................................................................................................................................................................................................................................................................................................
    408
  252. 222.
    Center of area
    ................................................................................................................................................................................................................................................................................................................................................................................................
    408
  253. 223.
    Moment of inertia. Plane areas
    ................................................................................................................................................................................................................................................................................................................................................................................................
    410
  254. 224.
    Polar moment of inertia. Rectangular coördinates
    ................................................................................................................................................................................................................................................................................................................................................................................................
    410
  255. 225.
    Polar moment of inertia. Polar coördinates
    ................................................................................................................................................................................................................................................................................................................................................................................................
    411
  256. 226.
    General method for finding the areas of surfaces
    ................................................................................................................................................................................................................................................................................................................................................................................................
    413
  257. 227.
    Volumes found by triple integration
    ................................................................................................................................................................................................................................................................................................................................................................................................
    417
  258.  

    CHAPTER XXX
    ORDINARY DIFFERENTIAL EQUATIONS

  259. 228.
    Differential equations. Order and degree
    ................................................................................................................................................................................................................................................................................................................................................................................................
    421
  260. 229.
    Solutions of differential equations
    ................................................................................................................................................................................................................................................................................................................................................................................................
    422
  261. 230.
    Verifications of solutions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    423
  262. 231.
    Differential equations of the first order and of the first degree
    ................................................................................................................................................................................................................................................................................................................................................................................................
    424
  263. 232.
    Differential equations of the \scriptstyle{n\text{th}} order and of the first degree
    ................................................................................................................................................................................................................................................................................................................................................................................................
    432
  264.  

    CHAPTER XXXI
    INTEGRAPH. APPROXIMATE INTEGRATION. TABLE OF INTEGRALS

  265. 233.
    Mechanical integration
    ................................................................................................................................................................................................................................................................................................................................................................................................
    443
  266. 234.
    Integral curves
    ................................................................................................................................................................................................................................................................................................................................................................................................
    443
  267. 235.
    The integraph
    ................................................................................................................................................................................................................................................................................................................................................................................................
    445
  268. 236.
    Polar planimeter
    ................................................................................................................................................................................................................................................................................................................................................................................................
    446
  269. 237.
    Area swept over by a line
    ................................................................................................................................................................................................................................................................................................................................................................................................
    446
  270. 238.
    Approximate integration
    ................................................................................................................................................................................................................................................................................................................................................................................................
    448
  271. 239.
    Trapezoidal rule
    ................................................................................................................................................................................................................................................................................................................................................................................................
    448
  272. 240.
    Simpson's rule (parabolic rule)
    ................................................................................................................................................................................................................................................................................................................................................................................................
    449
  273. 241.
    Integrals for reference
    ................................................................................................................................................................................................................................................................................................................................................................................................
    451
  274.  
  275. ................................................................................................................................................................................................................................................................................................................................................................................................
    461