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The Calculus Integral

Contents

Preface	i

Note to the instructor	iii

Table of Contents	vii

1 What you should know first	1
1.1 What is the calculus about?	1
1.2 What is an interval?	2
1.3 Sequences and series	4
1.4 Partitions	8
1.5 Continuous functions	10
1.6 Existence of maximum and minimum	21
1.7 Derivatives	23
1.8 Differentiation rules	25
1.9 Mean-value theorem	26
1.10 Lipschitz functions	33

2 The Indefinite Integral	35
2.1 An indefinite integral on an interval	35
2.2 Existence of indefinite integrals	39
2.3 Basic properties of indefinite integrals	4
2.3 Basic properties of indefinite integrals	42

3 The Definite Integral	49
3.1 Definition of the calculus integral	50
3.2 Integrability	55
3.3 Properties of the integral	58
3.4 Mean-value theorems for integrals	63
3.5 Riemann sums	64
3.6 Absolute integrability	79
3.7 Sequences and series of integrals	84
3.8 The monotone convergence theorem	97
3.9 Integration of power series	99
3.10 Applications of the integral	106
3.11 Numerical methods	114
3.12 More Exercises	119

4 Beyond the calculus integral	121
4.1 Countable sets	121
4.2 Derivatives which vanish outside of countable sets	123
4.3 Sets of measure zero	125
4.4 The Devil’s staircase	130
4.5 Functions with zero variation	132
4.6 The integral	138
4.7 Approximation by Riemann sums	142
4.8 Properties of the integral	143
4.9 The Henstock-Kurweil integral	148
4.10 The Riemann integral	149

5 Answers	153
5.1 Answers to problems	153

Index	288