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

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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 (${\displaystyle \scriptstyle {\infty }}$) ................................................................................................................................................................................................................................................................................................................................................................................................ 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 ${\displaystyle \scriptstyle {\frac {\sin x}{x}}}$ as ${\displaystyle \scriptstyle {x\doteq 0}}$ ................................................................................................................................................................................................................................................................................................................................................................................................ 21
27.  23 The number ${\displaystyle \scriptstyle {e}}$ ................................................................................................................................................................................................................................................................................................................................................................................................ 22
28.  24 Expressions assuming the form ${\displaystyle \scriptstyle {\frac {\infty }{\infty }}}$ ................................................................................................................................................................................................................................................................................................................................................................................................ 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 ${\displaystyle \scriptstyle {\sin v}}$ ................................................................................................................................................................................................................................................................................................................................................................................................ 54
56.  49 Differentiation of ${\displaystyle \scriptstyle {\cos v}}$ ................................................................................................................................................................................................................................................................................................................................................................................................ 55
57.  50 Differentiation of ${\displaystyle \scriptstyle {\tan v}}$ ................................................................................................................................................................................................................................................................................................................................................................................................ 56
58.  51 Differentiation of ${\displaystyle \scriptstyle {\cot v}}$ ................................................................................................................................................................................................................................................................................................................................................................................................ 56
59.  52 Differentiation of ${\displaystyle \scriptstyle {\sec v}}$ ................................................................................................................................................................................................................................................................................................................................................................................................ 56
60.  53 Differentiation of ${\displaystyle \scriptstyle {\csc v}}$ ................................................................................................................................................................................................................................................................................................................................................................................................ 57
61.  54 Differentiation of ${\displaystyle \scriptstyle {\operatorname {vers} v}}$ ................................................................................................................................................................................................................................................................................................................................................................................................ 57
62.  55 Differentiation of ${\displaystyle \scriptstyle {\operatorname {arc~sin} v}}$ ................................................................................................................................................................................................................................................................................................................................................................................................ 61
63.  56 Differentiation of ${\displaystyle \scriptstyle {\operatorname {arc~cos} v}}$ ................................................................................................................................................................................................................................................................................................................................................................................................ 62
64.  57 Differentiation of ${\displaystyle \scriptstyle {\operatorname {arc~tan} v}}$ ................................................................................................................................................................................................................................................................................................................................................................................................ 62
65.  58 Differentiation of ${\displaystyle \scriptstyle {\operatorname {arc~cot} v}}$ ................................................................................................................................................................................................................................................................................................................................................................................................ 63
66.  59 Differentiation of ${\displaystyle \scriptstyle {\operatorname {arc~sec} v}}$ ................................................................................................................................................................................................................................................................................................................................................................................................ 63
67.  60 Differentiation of ${\displaystyle \scriptstyle {\operatorname {arc~csc} v}}$ ................................................................................................................................................................................................................................................................................................................................................................................................ 64
68.  61 Differentiation of ${\displaystyle \scriptstyle {\operatorname {arc~vers} v}}$ ................................................................................................................................................................................................................................................................................................................................................................................................ 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 ${\displaystyle \scriptstyle {n{\text{th}}}}$ derivative ................................................................................................................................................................................................................................................................................................................................................................................................ 98
86.  77 Leibnitz's formula for the ${\displaystyle \scriptstyle {n{\text{th}}}}$ 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 ${\displaystyle \scriptstyle {\frac {0}{0}}}$ ................................................................................................................................................................................................................................................................................................................................................................................................ 171
127.  112 Evaluation of the indeterminate form ${\displaystyle \scriptstyle {\frac {\infty }{\infty }}}$ ................................................................................................................................................................................................................................................................................................................................................................................................ 174
128.  113 Evaluation of the indeterminate form ${\displaystyle \scriptstyle {0\cdot \infty }}$ ................................................................................................................................................................................................................................................................................................................................................................................................ 174
129.  114 Evaluation of the indeterminate form ${\displaystyle \scriptstyle {\infty -\infty }}$ ................................................................................................................................................................................................................................................................................................................................................................................................ 175
130.  115 Evaluation of the indeterminate forms ${\displaystyle \scriptstyle {0^{0},~1^{\infty },~\infty ^{0}}}$ ................................................................................................................................................................................................................................................................................................................................................................................................ 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 ${\displaystyle \scriptstyle {\sqrt {a^{2}-x^{2}}}}$ or ${\displaystyle \scriptstyle {\sqrt {x^{2}\pm a^{2}}}}$ 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 ${\displaystyle \scriptstyle {\phi (x)}}$ ................................................................................................................................................................................................................................................................................................................................................................................................ 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 ${\displaystyle \scriptstyle {y=\phi (x)}}$ 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 ${\displaystyle \scriptstyle {x}}$ only ................................................................................................................................................................................................................................................................................................................................................................................................ 335
218.  191 Differentials containing fractional powers of ${\displaystyle \scriptstyle {a+bx}}$ only ................................................................................................................................................................................................................................................................................................................................................................................................ 336
219.  192 Change in limits corresponding to change in variable ................................................................................................................................................................................................................................................................................................................................................................................................ 336
220.  193 Differentials containing no radical except ${\displaystyle \scriptstyle {\sqrt {a+bx+x^{2}}}}$ ................................................................................................................................................................................................................................................................................................................................................................................................ 338
221.  194 Differentials containing no radical except ${\displaystyle \scriptstyle {\sqrt {a+bx-x^{2}}}}$ ................................................................................................................................................................................................................................................................................................................................................................................................ 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 ${\displaystyle \scriptstyle {\int e^{ax}\sin {nx}dx}}$ and ${\displaystyle \scriptstyle {\int e^{ax}\cos {nx}dx}}$ ................................................................................................................................................................................................................................................................................................................................................................................................ 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 ${\displaystyle \scriptstyle {n{\text{th}}}}$ 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.  INDEX ................................................................................................................................................................................................................................................................................................................................................................................................ 461