# Elements of the Differential and Integral Calculus

 Elements of the Differential and Integral Calculus  (1911)  by William Anthony Granville
ELEMENTS OF THE DIFFERENTIAL
AND INTEGRAL CALCULUS
(REVISED EDITION)

BY

WILLIAM ANTHONY GRANVILLE, PH.D.. LL.D.

FORMERLY PRESIDENT OF PENNSYLYANIA COLLEGE

WITH THE EDITORIAL COOPERATION OF

PERCEY F. SMITH, PH.D.

PROFESSOR OF MATHEMATICS IN THE SHEFFIELD SCIENTIFIC SCHOOL
YALE UNIVERSITY

## Contents

### Differential Calculus

CHAPTER I
COLLECTION OF FORMULAS

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

CHAPTER II
VARIABLES AND FUNCTIONS

1. Variables and constants
2. Interval of a variable
3. Continuous variation
4. Functions
5. Independent and dependent variables
6. Notation of functions
7. Values of the independent variable for which a function is defined

CHAPTER III
THEORY OF LIMITS

1. Limit of a variable
2. Division by zero excluded
3. Infinitesimals
4. The concept of infinity ($\infty$)
5. Limiting value of a function
6. Continuous and discontinuous functions
7. Continuity and discontinuity of functions illustrated by their graphs
8. Fundamental theorems on limits
9. Special limiting values
10. The limit of $\tfrac{\sin\ x}{x}$ as $x \dot= 0$
11. The number $e$
12. Expressions assuming the form $\tfrac{\infty}{\infty}$

CHAPTER IV
DIFFERENTIATION

CHAPTER V
RULES FOR DIFFERENTIATING STANDARD ELEMENTARY FORM

1. Importance of General Rule
2. Differentiation of a constant
3. Differentiation of a variable with respect to itself
4. Differentiation of a sum
5. Differentiation of the product of a constant and a function
6. Differentiation of the product of two functions
7. Differentiation of the product of any finite number of functions
8. Differentiation of a function with a constant exponent
9. Differentiation of a quotient
10. Differentiation of a function of a function
11. Differentiation of inverse functions
12. Differentiation of a logarithm
13. Differentiation of the simple exponential function
14. Differentiation of the general exponential function
15. Logarithmic differentiation
16. Differentiation of $\sin\ v$
17. Differentiation of $\cos\ v$
18. Differentiation of $\tan\ v$
19. Differentiation of $\cot\ v$
20. Differentiation of $\sec\ v$
21. Differentiation of $\csc\ v$
22. Differentiation of $\operatorname{vers}\ v$[1]
23. Differentiation of $\arcsin\ v$
24. Differentiation of $\arccos\ v$
25. Differentiation of $\arctan\ v$
26. Differentiation of $\arccot\ v$
27. Differentiation of $\arcsec\ v$
28. Differentiation of $\arccsc\ v$
29. Differentiation of $\operatorname{arcvers}\ v$
30. Implicit functions
31. Differentiation of implicit functions

CHAPTER VI
SIMPLE APPLICATIONS OF THE DERIVATIVE

1. Direction of a curve
2. Equations of tangent and normal, lengths of subtangent and subnormal
3. Rectangular coördinates
4. Parametric equations of a curve
5. Angle between the radius vector drawn to a point on a curve and the tangent to the curve at that point
6. Lengths of polar subtangent and polar subnormal
7. Solution of equations having multiple roots
8. Applications of the derivative in mechanics. Velocity
9. Component velocities
10. Acceleration
11. Component accelerations

CHAPTER VII
SUCCESSIVE DIFFERENTIATION

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

1. Introduction
2. Increasing and decreasing functions
3. Tests for determining when a function is increasing and when decreasing
4. Maximum and minimum values of a function
5. First method for examining a function for maximum and minimum values
6. Second method for examining a function for maximum and minimum values
7. Definition of points of inflection and rule for finding points of inflection
8. Curve tracing

CHAPTER IX
DIFFERENTIALS

CHAPTER X
RATES

CHAPTER XI
CHANGE OF VARIABLE

CHAPTER XII

CHAPTER XIII
THEOREM OF MEAN VALUE. INDETERMINATE FORMS

CHAPTER XIV
CIRCLE OF CURVATURE. CENTER OF CURVATURE

CHAPTER XV
PARTIAL DIFFERENTIATION

CHAPTER XVI
ENVELOPES

CHAPTER XVII
SERIES

CHAPTER XVIII
EXPANSION OF FUNCTIONS

CHAPTER XIX
ASYMPTOTES. SINGULAR POINTS

1. Rectilinear asymptotes
2. Asymptotes found by method of limiting intercepts
3. Method of determining asymptotes to algebraic curves
4. Asymptotes in polar coördinates
5. Singular points
6. Determination of the tangent to an algebraic curve at a given point by inspection
7. Nodes
8. Cusps
9. Conjugate or isolated points
10. Transcendental singularities

CHAPTER XX
APPLICATIONS TO GEOMETRY OF SPACE

1. Tangent line and normal plane to a skew curve whose equations are given in parametric form
2. Tangent plane to a surface
3. Normal line to a surface
4. Another form of the equations of the tangent line to a skew curve
5. Another form of the equation of the normal plane to a skew curve

CHAPTER XXI
CURVES FOR REFERENCE

### Integral Calculus

CHAPTER XXII
INTEGRATION. RULES FOR INTEGRATING STANDARD ELEMENTARY FORMS

1. Integration
2. Constant of integration. Indefinite integral
3. Rules for integrating standard elementary forms
4. Trigonometric differentials
5. Integration of expressions containing $\sqrt{a^2-x^2}$ or $\sqrt{x^2 \pm a^2}$ by a trigonometric substitution

CHAPTER XXIII
CONSTANT OF INTEGRATION

1. Determination of the constant of integration by means of initial conditions
2. Geometrical signification of the constant of integration
3. Physical signification of the constant of integration

CHAPTER XXIV
THE DEFINITE INTEGRAL

1. Differential of an area
2. The definite integral
3. Calculation of a definite integral
4. Calculation of areas
5. Geometrical representation of an integral
6. Mean value of $\Phi(x)$
7. Interchange of limits
8. Decomposition of the interval
9. The definite integral a function of its limits
10. Infinite limits
11. When $y = \Phi(x)$ is discontinuous

CHAPTER XXV
INTEGRATION OF RATIONAL FRACTIONS

1. Introduction
2. Case I
3. Case II
4. Case III
5. Case IV

CHAPTER XXVI
INTEGRATION BY SUBSTITUTION OF A NEW VARIABLE. RATIONALIZATION

1. Introduction
2. Differentials containing fractional powers of $x$ only
3. Differentials containing fractional powers of $a + bx$ only
4. Change in limits corresponding to change in variable
5. Differentials containing no radical except $\sqrt{a + bx + x^2}$
6. Differentials containing no radical except $\sqrt{a + bx - x^2}$
7. Binomial differentials
8. Conditions of integrability of binomial differentials
9. Transformation of trigonometric differentials
10. Miscellaneous substitutions

CHAPTER XXVII
INTEGRATION BY PARTS. REDUCTION FORMULAS

1. Formula for integration by parts
2. Reduction formulas for binomial differentials
3. Reduction formulas for trigonometric differentials
4. To find $\int e^{ax}\sin{nx}dx$ and $\int e^{ax} \cos{nx}dx$

CHAPTER XXVIII
INTEGRATION A PROCESS OF SUMMATION

1. Introduction
2. The fundamental theorem of Integral Calculus
3. Analytical proof of the Fundamental Theorem
4. Areas of plane curves. Rectangular coördinates
5. Area when curve is given in parametric form
6. Areas of plane curves. Polar coördinates
7. Length of a curve
8. Lengths of plane curves. Rectangular coördinates
9. Lengths of plane curves. Polar coördinates
10. Volumes of solids of revolution
11. Areas of surfaces of revolution
12. Miscellaneous applications

CHAPTER XXIX
SUCCESSIVE AND PARTIAL INTEGRATION

1. Successive integration
2. Partial integration
3. Definite double integral. Geometric interpretation
4. Value of a definite double integral over a region
5. Plane area as a definite double integral. Rectangular coördinates
6. Plane area as a definite double integral. Polar coördinates
7. Moment of area
8. Center of area
9. Moment of inertia. Plane areas
10. Polar moment of inertia. Rectangular coördinates
11. Polar moment of inertia. Polar coördinates
12. General method for finding the areas of surfaces
13. Volumes found by triple integration

CHAPTER XXX
ORDINARY DIFFERENTIAL EQUATIONS

1. Differential equations. Order and degree
2. Solutions of differential equations
3. Verifications of solutions
4. Differential equations of the first order and of the first degree
5. Differential equations of the nth order and of the first degree

CHAPTER XXXI
INTEGRAPH. APPROXIMATE INTEGRATION. TABLE OF INTEGRALS

1. Mechanical integration
2. Integral curves
3. The integraph
4. Polar planimeter
5. Area swept over by a line
6. Approximate integration
7. Trapezoidal rule
8. Simpson's rule (parabolic rule)
9. Integrals for reference

## Notes

1. The function vers is an abbreviation for versine, which is equal to $1 - \cos$.

 This work is in the public domain in the United States because it was published before January 1, 1923. The author died in 1943, so this work is also in the public domain in countries and areas where the copyright term is the author's life plus 70 years or less. This work may also be in the public domain in countries and areas with longer native copyright terms that apply the rule of the shorter term to foreign works.