# Book:R.P. Gillespie/Integration/Second Edition

## R.P. Gillespie: Integration (2nd Edition)

Published $\text {1944}$, Oliver & Boyd

### Contents

Preface
Preface to the Second Edition
chapter $\text {I}$: INTRODUCTION
1. Area of a Circle
2. Areas
3. The Integral as the Limit of a Sum
4. Volume of Revolution
5. Properties of the Integral
chapter $\text {II}$: INTEGRATION OF ELEMENTARY FUNCTIONS
6. Methods of Integration
7. Standard Integrals
8. Change of Variable
9. Integration by Parts
10. Integration of Rational Functions
11. Integration of Trigonometric Functions
12. Integration of Irrational Functions
Examples $\textit {I}$
chapter $\text {III}$: MULTIPLE INTEGRALS
13. Double Integrals
14. Repeated Integrals
15. Triple Integrals
16. Change of Variable in Multiple Integrals
17. Polar Coordinates
Examples $\textit {II}$
chapter $\text {IV}$: CURVILINEAR AND SURFACE INTEGRALS
18. Length of a Curve
19. Curvilinear or Line Integrals
20. Theorem
21. Surface of Revolution
22. The Element of Surface
23. Freedom Equations of a Surface
24. Surface Integrals
25. Gauss's Theorem
Examples $\textit {III}$
chapter $\text {V}$: THE RIEMANN INTEGRAL
27. Preliminary Definitions
28. Definition of the Integral
29. Fundamental Theorem
30. Integration of Continuous Functions
31. Properties of the Integral
chapter $\text {VI}$: INFINITE INTEGRALS
32. Definition
33. Fundamental Theorem
34. Absolute Convergence
35. Infinite Integrand
36. Two Important Integrals
37. Gamma and Beta Functions
38. Relation between Gamma and Beta Functions
39. Uniform Convergence of an Integral
40. Theorem on Repeated Integrals
41. Differentiation under the Integral Sign
Examples $\textit {IV}$
chapter $\text {VII}$: THE RIEMANN DOUBLE INTEGRAL
43. Change of Variable
44. Integration by Parts
45. Theorems of Mean Value
46. Integration of Series
47. Orthogonal Systems of Functions
48. Differentiation under the Integral Sign
49. Double Integral over a Rectangle
50. Rectifiable Curves
51. Theorem on Rectifiable Curves
52. Double Integrals
53. Repeated Integrals
54. The Generalised Element of Area
Examples $\textit {V}$
Index

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## Source work progress

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