An elementary introduction to integration theory on the real line. This is at the level of an honor's course in calculus or a first undergraduate level real analysis course. In the end the student should be adequately prepared for a graduate level course in Lebesgue integration.
Conditions of Use
This book is licensed under a Creative Commons License (CC BY-NC-SA). You can download the ebook The Calculus Integral for free.
- Title
- The Calculus Integral
- Author(s)
- Brian S Thomson
- Published
- 2010-05-27
- Edition
- 1
- Format
- eBook (pdf, epub, mobi)
- Pages
- 304
- Language
- English
- ISBN-10
- 1442180951
- ISBN-13
- 9781442180956
- License
- CC BY-NC-SA
- Book Homepage
- Free eBook, Errata, Code, Solutions, etc.
1 What you should know first 1.1 What is the calculus about? 1.2 What is an interval? 1.3 Sequences and series 1.4 Partitions 1.5 Continuous functions 1.6 Existence of maximum and minimum 1.7 Derivatives 1.8 Differentiation rules 1.9 Mean-value theorem 1.10 Lipschitz functions 2 The Indefinite Integral 2.1 An indefinite integral on an interval 2.2 Existence of indefinite integrals 2.3 Basic properties of indefinite integrals 2.3 Basic properties of indefinite integrals 3 The Definite Integral 3.1 Definition of the calculus integral 3.2 Integrability 3.3 Properties of the integral 3.4 Mean-value theorems for integrals 3.5 Riemann sums 3.6 Absolute integrability 3.7 Sequences and series of integrals 3.8 The monotone convergence theorem 3.9 Integration of power series 3.10 Applications of the integral 3.11 Numerical methods 3.12 More Exercises 4 Beyond the calculus integral 4.1 Countable sets 4.2 Derivatives which vanish outside of countable sets 4.3 Sets of measure zero 4.4 The Devil’s staircase 4.5 Functions with zero variation 4.6 The integral 4.7 Approximation by Riemann sums 4.8 Properties of the integral 4.9 The Henstock-Kurweil integral 4.10 The Riemann integral 5 Answers 5.1 Answers to problems Index
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