Calculus Volume 2

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.

Please note that this title is published under a CC BY-NC-SA 4.0 license, which means that you are free to use and adapt, but not for commercial purposes. Changes you make need to be shared using this license.

Senior Contributing Authors

Gilbert Strang, Massachusetts Institute of Technology
Edwin “Jed” Herman, University of Wisconsin-Stevens Point


Contributing Authors

William Radulovich, Formerly at Florida State College at Jacksonville
Erica M. Rutter, Arizona State University
David Smith, University of the Virgin Islands
Kirsten R. Messer, Colorado State University-Pueblo
Alfred K. Mulzet, Florida State College at Jacksonville
Nicoleta Virginia Bila, Fayetteville State University
Sheri J. Boyd, Rollins College
Joyati Debnath, Winona State University
Valeree Falduto, Palm Beach State College
Elaine A. Terry, Saint Joseph’s University
David Torain, Hampton University
Michelle Merriweather, Bronxville High School
Catherine Abbott, Keuka College
Joseph Lakey, New Mexico State University
Julie Levandosky, Framingham State University
David McCune, William Jewell College

Chapter 1 Integration
• Introduction
• 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 Review
o Key Terms
o Key Equations
o Key Concepts
o Review Exercises

Chapter 2 Applications of Integration
• Introduction
• 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 Review
o Key Terms
o Key Equations
o Key Concepts
o Review Exercises

Chapter 3 Techniques of Integration
• Introduction
• 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 Review
o Key Terms
o Key Equations
o Key Concepts
o Review Exercises

Chapter 4 Introduction to Differential Equations
• Introduction
• 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 Review
o Key Terms
o Key Equations
o Key Concepts
o Review Exercises

Chapter 5 Sequences and Series
• Introduction
• 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 Review
o Key Terms
o Key Equations
o Key Concepts
o Review Exercises

Chapter 6 Power Series
• Introduction
• 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 Review
o Key Terms
o Key Equations
o Key Concepts
o Review Exercises

Chapter 7 Parametric Equations and Polar Coordinates
• Introduction
• 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
Chapter Review
o Key Terms
o Key Equations
o Key Concepts
o Review Exercises