Elements of the Differential and Integral Calculus/Contents

by William Anthony Granville

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 $\scriptstyle {x\doteq 0}$ ................................................................................................................................................................................................................................................................................................................................................................................................

    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. INDEX ................................................................................................................................................................................................................................................................................................................................................................................................

     461