Notes on
Calculus II
Integral Calculus
Miguel A. Lerma
,November 22, 2002
, Contents
Introduction 5
Chapter 1. Integrals 6
1.1. Areas and Distances. The Definite Integral 6
1.2. The Evaluation Theorem 11
1.3. The Fundamental Theorem of Calculus 14
1.4. The Substitution Rule 16
1.5. Integration by Parts 21
1.6. Trigonometric Integrals and Trigonometric Substitutions 26
1.7. Partial Fractions 32
1.8. Integration using Tables and CAS 39
1.9. Numerical Integration 41
1.10. Improper Integrals 46
Chapter 2. Applications of Integration 50
2.1. More about Areas 50
2.2. Volumes 52
2.3. Arc Length, Parametric Curves 57
2.4. Average Value of a Function (Mean Value Theorem) 61
2.5. Applications to Physics and Engineering 63
2.6. Probability 69
Chapter 3. Differential Equations 74
3.1. Differential Equations and Separable Equations 74
3.2. Directional Fields and Euler’s Method 78
3.3. Exponential Growth and Decay 80
Chapter 4. Infinite Sequences and Series 83
4.1. Sequences 83
4.2. Series 88
4.3. The Integral and Comparison Tests 92
4.4. Other Convergence Tests 96
4.5. Power Series 98
4.6. Representation of Functions as Power Series 100
4.7. Taylor and MacLaurin Series 103
3
, CONTENTS 4
4.8. Applications of Taylor Polynomials 109
Appendix A. Hyperbolic Functions 113
A.1. Hyperbolic Functions 113
Appendix B. Various Formulas 118
B.1. Summation Formulas 118
Appendix C. Table of Integrals 119
Calculus II
Integral Calculus
Miguel A. Lerma
,November 22, 2002
, Contents
Introduction 5
Chapter 1. Integrals 6
1.1. Areas and Distances. The Definite Integral 6
1.2. The Evaluation Theorem 11
1.3. The Fundamental Theorem of Calculus 14
1.4. The Substitution Rule 16
1.5. Integration by Parts 21
1.6. Trigonometric Integrals and Trigonometric Substitutions 26
1.7. Partial Fractions 32
1.8. Integration using Tables and CAS 39
1.9. Numerical Integration 41
1.10. Improper Integrals 46
Chapter 2. Applications of Integration 50
2.1. More about Areas 50
2.2. Volumes 52
2.3. Arc Length, Parametric Curves 57
2.4. Average Value of a Function (Mean Value Theorem) 61
2.5. Applications to Physics and Engineering 63
2.6. Probability 69
Chapter 3. Differential Equations 74
3.1. Differential Equations and Separable Equations 74
3.2. Directional Fields and Euler’s Method 78
3.3. Exponential Growth and Decay 80
Chapter 4. Infinite Sequences and Series 83
4.1. Sequences 83
4.2. Series 88
4.3. The Integral and Comparison Tests 92
4.4. Other Convergence Tests 96
4.5. Power Series 98
4.6. Representation of Functions as Power Series 100
4.7. Taylor and MacLaurin Series 103
3
, CONTENTS 4
4.8. Applications of Taylor Polynomials 109
Appendix A. Hyperbolic Functions 113
A.1. Hyperbolic Functions 113
Appendix B. Various Formulas 118
B.1. Summation Formulas 118
Appendix C. Table of Integrals 119
