# Calculus Volume 2

Gilbert Strang, Massachusetts Institute of Technology

Pub Date: 2016

ISBN 13: 978-1-9381680-6-2

Publisher: OpenStax

## Conditions of Use

Attribution-NonCommercial-ShareAlike

CC BY-NC-SA

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## Table of Contents

Preface

**Chapter 1: Integration**

- 1.1 Approximating Areas
- 1.2 The Definite Integral
- 1.3 The Fundamental Theorem of Calculus
- 1.4 Integration Formulas and the Net Change Theorem
- 1.5 Substitution
- 1.6 Integrals Involving Exponential and Logarithmic Functions
- 1.7 Integrals Resulting in Inverse Trigonometric Functions

**Chapter 2: Applications of Integration**

- 2.1 Areas between Curves
- 2.2 Determining Volumes by Slicing
- 2.3 Volumes of Revolution: Cylindrical Shells
- 2.4 Arc Length of a Curve and Surface Area
- 2.5 Physical Applications
- 2.6 Moments and Centers of Mass
- 2.7 Integrals, Exponential Functions, and Logarithms
- 2.8 Exponential Growth and Decay
- 2.9 Calculus of the Hyperbolic Functions

**Chapter 3: Techniques of Integration **

- 3.1 Integration by Parts
- 3.2 Trigonometric Integrals
- 3.3 Trigonometric Substitution
- 3.4 Partial Fractions
- 3.5 Other Strategies for Integration
- 3.6 Numerical Integration
- 3.7 Improper Integrals

**Chapter 4: Introduction to Differential Equations **

- 4.1 Basics of Differential Equations
- 4.2 Direction Fields and Numerical Methods
- 4.3 Separable Equations
- 4.4 The Logistic Equation
- 4.5 First-order Linear Equations

**Chapter 5: Sequences and Series**

- 5.1 Sequences
- 5.2 Infinite Series
- 5.3 The Divergence and Integral Tests
- 5.4 Comparison Tests
- 5.5 Alternating Series
- 5.6 Ratio and Root Tests

**Chapter 6: Power Series **

- 6.1 Power Series and Functions
- 6.2 Properties of Power Series
- 6.3 Taylor and Maclaurin Series
- 6.4 Working with Taylor Series

**Chapter 7: Parametric Equations and Polar Coordinates **

- 7.1 Parametric Equations
- 7.2 Calculus of Parametric Curves
- 7.3 Polar Coordinates
- 7.4 Area and Arc Length in Polar Coordinates
- 7.5 Conic Sections

Appendix A: Table of Integrals

Appendix B: Table of Derivatives

Appendix C: Review of Pre-Calculus

Index

## About the Book

*Calculus *is designed for the typical two- or three-semester general calculus course, incorporating innovative features to enhance student learning. The book guides students through the core concepts of calculus and helps them understand how those concepts apply to their lives and the world around them. Due to the comprehensive nature of the material, we are offering the book in three volumes for flexibility and efficiency. Volume 2 covers integration, differential equations, sequences and series, and parametric equations and polar coordinates.

## About the Contributors

### Author(s)

**Gilbert Strang** was an undergraduate at MIT and a Rhodes Scholar at Balliol College, Oxford. His Ph.D. was from UCLA and since then he has taught at MIT. He has been a Sloan Fellow and a Fairchild Scholar and is a Fellow of the American Academy of Arts and Sciences. He is a Professor of Mathematics at MIT, an Honorary Fellow of Balliol College, and a member of the National Academy of Sciences.

He was the President of SIAM during 1999 and 2000, and Chair of the Joint Policy Board for Mathematics. He received the von Neumann Medal of the US Association for Computational Mechanics, and the Henrici Prize for applied analysis. The first Su Buchin Prize from the International Congress of Industrial and Applied Mathematics, and the Haimo Prize from the Mathematical Association of America, were awarded for his contributions to teaching around the world.