Skip to ContentGo to accessibility pageKeyboard shortcuts menu
OpenStax Logo

Welcome to Calculus Volume 2, an OpenStax resource. This textbook was written to increase student access to high-quality learning materials, maintaining highest standards of academic rigor at little to no cost.

About OpenStax

OpenStax is a nonprofit based at Rice University, and it’s our mission to improve student access to education. Our first openly licensed college textbook was published in 2012, and our library has since scaled to over 20 books for college and AP courses used by hundreds of thousands of students. Our adaptive learning technology, designed to improve learning outcomes through personalized educational paths, is being piloted in college courses throughout the country. Through our partnerships with philanthropic foundations and our alliance with other educational resource organizations, OpenStax is breaking down the most common barriers to learning and empowering students and instructors to succeed.

About OpenStax's Resources


Calculus Volume 2 is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC-BY-NC-SA) license, which means that you can distribute, remix, and build upon the content, as long as you provide attribution to OpenStax and its content contributors, do not use the content for commercial purposes, and distribute the content under the same CC-BY-NC-SA license.

Because our books are openly licensed, you are free to use the entire book or pick and choose the sections that are most relevant to the needs of your course. Feel free to remix the content by assigning your students certain chapters and sections in your syllabus, in the order that you prefer. You can even provide a direct link in your syllabus to the sections in the web view of your book.

Instructors also have the option of creating a customized version of their OpenStax book. The custom version can be made available to students in low-cost print or digital form through their campus bookstore. Visit your book page on for more information.


All OpenStax textbooks undergo a rigorous review process. However, like any professional-grade textbook, errors sometimes occur. Since our books are web based, we can make updates periodically when deemed pedagogically necessary. If you have a correction to suggest, submit it through the link on your book page on Subject matter experts review all errata suggestions. OpenStax is committed to remaining transparent about all updates, so you will also find a list of past errata changes on your book page on


You can access this textbook for free in web view or PDF through, and for a low cost in print.

About 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.

Coverage and Scope

Our Calculus Volume 2 textbook adheres to the scope and sequence of most general calculus courses nationwide. We have worked to make calculus interesting and accessible to students while maintaining the mathematical rigor inherent in the subject. With this objective in mind, the content of the three volumes of Calculus have been developed and arranged to provide a logical progression from fundamental to more advanced concepts, building upon what students have already learned and emphasizing connections between topics and between theory and applications. The goal of each section is to enable students not just to recognize concepts, but work with them in ways that will be useful in later courses and future careers. The organization and pedagogical features were developed and vetted with feedback from mathematics educators dedicated to the project.

Volume 1
  • Chapter 1: Functions and Graphs
  • Chapter 2: Limits
  • Chapter 3: Derivatives
  • Chapter 4: Applications of Derivatives
  • Chapter 5: Integration
  • Chapter 6: Applications of Integration
Volume 2
  • Chapter 1: Integration
  • Chapter 2: Applications of Integration
  • Chapter 3: Techniques of Integration
  • Chapter 4: Introduction to Differential Equations
  • Chapter 5: Sequences and Series
  • Chapter 6: Power Series
  • Chapter 7: Parametric Equations and Polar Coordinates
Volume 3
  • Chapter 1: Parametric Equations and Polar Coordinates
  • Chapter 2: Vectors in Space
  • Chapter 3: Vector-Valued Functions
  • Chapter 4: Differentiation of Functions of Several Variables
  • Chapter 5: Multiple Integration
  • Chapter 6: Vector Calculus
  • Chapter 7: Second-Order Differential Equations

Pedagogical Foundation

Throughout Calculus Volume 2 you will find examples and exercises that present classical ideas and techniques as well as modern applications and methods. Derivations and explanations are based on years of classroom experience on the part of long-time calculus professors, striving for a balance of clarity and rigor that has proven successful with their students. Motivational applications cover important topics in probability, biology, ecology, business, and economics, as well as areas of physics, chemistry, engineering, and computer science. Student Projects in each chapter give students opportunities to explore interesting sidelights in pure and applied mathematics, from showing that the number e is irrational, to calculating the center of mass of the Grand Canyon Skywalk or the terminal speed of a skydiver. Chapter Opening Applications pose problems that are solved later in the chapter, using the ideas covered in that chapter. Problems include the hydraulic force against the Hoover Dam, and the comparison of the relative intensity of two earthquakes. Definitions, Rules, and Theorems are highlighted throughout the text, including over 60 Proofs of theorems.

Assessments That Reinforce Key Concepts

In-chapter Examples walk students through problems by posing a question, stepping out a solution, and then asking students to practice the skill with a “Check Your Learning” component. The book also includes assessments at the end of each chapter so students can apply what they’ve learned through practice problems. Many exercises are marked with a [T] to indicate they are suitable for solution by technology, including calculators or Computer Algebra Systems (CAS). Answers for selected exercises are available in the Answer Key at the back of the book.

Early or Late Transcendentals

Calculus Volume 2 is designed to accommodate both Early and Late Transcendental approaches to calculus. Exponential and logarithmic functions are presented in Chapter 2. Integration of these functions is covered in Chapters 1 for instructors who want to include them with other types of functions. These discussions, however, are in separate sections that can be skipped for instructors who prefer to wait until the integral definitions are given before teaching the calculus derivations of exponentials and logarithms.

Comprehensive Art Program

Our art program is designed to enhance students’ understanding of concepts through clear and effective illustrations, diagrams, and photographs.

... ...

Answers to Questions in the Book

Answers to Examples are provided just below the question in the book. All Checkpoint answers are provided in the Answer Key. Odd-numbered Exercises and Review Exercises questions are provided to students in the Answer Key as well as the Student Solution Guide on the Student Resources page. Even-numbered answers are provided only to instructors in the Instructor Answer Guide via the Instructor Resources page.

Additional Resources

Student and Instructor Resources

We’ve compiled additional resources for both students and instructors, including Getting Started Guides, an instructor solution manual, and PowerPoint slides. Instructor resources require a verified instructor account, which can be requested on your log-in. Take advantage of these resources to supplement your OpenStax book.

Partner Resources

OpenStax Partners are our allies in the mission to make high-quality learning materials affordable and accessible to students and instructors everywhere. Their tools integrate seamlessly with our OpenStax titles at a low cost. To access the partner resources for your text, visit your book page on

About The Authors

Senior Contributing Authors

Gilbert Strang, Massachusetts Institute of Technology
Dr. Strang received his PhD from UCLA in 1959 and has been teaching mathematics at MIT ever since. His Calculus online textbook is one of eleven that he has published and is the basis from which our final product has been derived and updated for today’s student. Strang is a decorated mathematician and past Rhodes Scholar at Oxford University.

Edwin “Jed” Herman, University of Wisconsin-Stevens Point
Dr. Herman earned a BS in Mathematics from Harvey Mudd College in 1985, an MA in Mathematics from UCLA in 1987, and a PhD in Mathematics from the University of Oregon in 1997. He is currently a Professor at the University of Wisconsin-Stevens Point. He has more than 20 years of experience teaching college mathematics, is a student research mentor, is experienced in course development/design, and is also an avid board game designer and player.

Contributing Authors

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


Marwan A. Abu-Sawwa, Florida State College at Jacksonville
Kenneth J. Bernard, Virginia State University
John Beyers, University of Maryland
Charles Buehrle, Franklin & Marshall College
Matthew Cathey, Wofford College
Michael Cohen, Hofstra University
William DeSalazar, Broward County School System
Murray Eisenberg, University of Massachusetts Amherst
Kristyanna Erickson, Cecil College
Tiernan Fogarty, Oregon Institute of Technology
David French, Tidewater Community College
Marilyn Gloyer, Virginia Commonwealth University
Shawna Haider, Salt Lake Community College
Lance Hemlow, Raritan Valley Community College
Jerry Jared, The Blue Ridge School
Peter Jipsen, Chapman University
David Johnson, Lehigh University
M.R. Khadivi, Jackson State University
Robert J. Krueger, Concordia University
Tor A. Kwembe, Jackson State University
Jean-Marie Magnier, Springfield Technical Community College
Cheryl Chute Miller, SUNY Potsdam
Bagisa Mukherjee, Penn State University, Worthington Scranton Campus
Kasso Okoudjou, University of Maryland College Park
Peter Olszewski, Penn State Erie, The Behrend College
Steven Purtee, Valencia College
Alice Ramos, Bethel College
Doug Shaw, University of Northern Iowa
Hussain Elalaoui-Talibi, Tuskegee University
Jeffrey Taub, Maine Maritime Academy
William Thistleton, SUNY Polytechnic Institute
A. David Trubatch, Montclair State University
Carmen Wright, Jackson State University
Zhenbu Zhang, Jackson State University

Order a print copy

As an Amazon Associate we earn from qualifying purchases.


This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution-NonCommercial-ShareAlike License and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at
Citation information

© Feb 5, 2024 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.