Calculus, Volume 2

829 Pages·2017·33.98 MB·english

by  Gilbert Strang

Volume 2

Calculus Volume 2

SENIOR CONTRIBUTING AUTHORS
EDWIN "JED" HERMAN, UNIVERSITY OF WISCONSIN-STEVENS POINT
GILBERT STRANG, MASSACHUSETTS INSTITUTE OF TECHNOLOGY
OpenStax
Rice University
6100 Main Street MS-375
Houston, Texas 77005

To learn more about OpenStax, visit https://openstax.org.
Individual print copies and bulk orders can be purchased through our website.

©2017 Rice University. Textbook content produced by OpenStax is licensed under a Creative Commons
Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0). Under this license, any
user of this textbook or the textbook contents herein can share, remix, and build upon the content for
noncommercial purposes only. Any adaptations must be shared under the same type of license. In any case of
sharing the original or adapted material, whether in whole or in part, the user must provide proper attribution
as follows:

- If you noncommercially redistribute this textbook in a digital format (including but not limited to PDF and
HTML), then you must retain on every page the following attribution:
“Download for free at https://openstax.org/details/books/calculus-volume-2.”
- If you noncommercially redistribute this textbook in a print format, then you must include on every
physical page the following attribution:
“Download for free at https://openstax.org/details/books/calculus-volume-2.”
- If you noncommercially redistribute part of this textbook, then you must retain in every digital format
page view (including but not limited to PDF and HTML) and on every physical printed page the following
attribution:
“Download for free at https://openstax.org/details/books/calculus-volume-2.”
- If you use this textbook as a bibliographic reference, please include
https://openstax.org/details/books/calculus-volume-2 in your citation.

For questions regarding this licensing, please contact [email protected].

Trademarks
The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, OpenStax CNX logo,
OpenStax Tutor name, Openstax Tutor logo, Connexions name, Connexions logo, Rice University name, and
Rice University logo are not subject to the license and may not be reproduced without the prior and express
written consent of Rice University.

PRINT BOOK ISBN-10 1-938168-06-2
PRINT BOOK ISBN-13 978-1-938168-06-2
PDF VERSION ISBN-10 1-947172-14-X
PDF VERSION ISBN-13 978-1-947172-14-2
Revision Number C2-2016-002(03/17)-LC
Original Publication Year 2016
OPENSTAX
OpenStax provides free, peer-reviewed, openly licensed textbooks for introductory college and Advanced Placement®
courses and low-cost, personalized courseware that helps students learn. A nonprofit ed tech initiative based at Rice
University, we’re committed to helping students access the tools they need to complete their courses and meet their
educational goals.

RICE UNIVERSITY
OpenStax, OpenStax CNX, and OpenStax Tutor are initiatives of Rice University. As a leading research university with a
distinctive commitment to undergraduate education, Rice University aspires to path-breaking research, unsurpassed
teaching, and contributions to the betterment of our world. It seeks to fulfill this mission by cultivating a diverse community
of learning and discovery that produces leaders across the spectrum of human endeavor.

FOUNDATION SUPPORT

OpenStax is grateful for the tremendous support of our sponsors. Without their strong engagement, the goal
of free access to high-quality textbooks would remain just a dream.

Laura and John Arnold Foundation (LJAF) actively seeks opportunities to invest in organizations and
thought leaders that have a sincere interest in implementing fundamental changes that not only
yield immediate gains, but also repair broken systems for future generations. LJAF currently focuses
its strategic investments on education, criminal justice, research integrity, and public accountability.

The William and Flora Hewlett Foundation has been making grants since 1967 to help solve social
and environmental problems at home and around the world. The Foundation concentrates its
resources on activities in education, the environment, global development and population,
performing arts, and philanthropy, and makes grants to support disadvantaged communities in the
San Francisco Bay Area.

Calvin K. Kazanjian was the founder and president of Peter Paul (Almond Joy), Inc. He firmly believed
that the more people understood about basic economics the happier and more prosperous they
would be. Accordingly, he established the Calvin K. Kazanjian Economics Foundation Inc, in 1949 as a
philanthropic, nonpolitical educational organization to support efforts that enhanced economic
understanding.

Guided by the belief that every life has equal value, the Bill & Melinda Gates Foundation works to
help all people lead healthy, productive lives. In developing countries, it focuses on improving
people’s health with vaccines and other life-saving tools and giving them the chance to lift
themselves out of hunger and extreme poverty. In the United States, it seeks to significantly
improve education so that all young people have the opportunity to reach their full potential. Based
in Seattle, Washington, the foundation is led by CEO Jeff Raikes and Co-chair William H. Gates Sr.,
under the direction of Bill and Melinda Gates and Warren Buffett.

The Maxfield Foundation supports projects with potential for high impact in science, education,
sustainability, and other areas of social importance.

Our mission at The Michelson 20MM Foundation is to grow access and success by eliminating
unnecessary hurdles to affordability. We support the creation, sharing, and proliferation of more
effective, more affordable educational content by leveraging disruptive technologies, open
educational resources, and new models for collaboration between for-profit, nonprofit, and public
entities.

The Bill and Stephanie Sick Fund supports innovative projects in the areas of Education, Art, Science
and Engineering.
IT’S INNOVATION IN EDUCATION. A
HII W O U L D N ’ T T H I S PENSTAX
CII LOOK BETTER TUDENTS FREE
TE ON A BR AND MEET SCOPE AND
SE NEW IPAD QUIREMENTS FOR
MII MINI? URSES. THESE ARE
PEER-REVIEWED TEXTS WRITTEN BY
Knowing where our textbooks are used can
PROFESSIOhelp Nus prAovidLe be tCter seOrviceNs to TstudEentNs andT A
receive more grant support for future projects.
DEVELOPERS. ADOPT A BOOK
If you’re using an OpenStax textbook, either as
required for your course or just as an
extra resource, send your course syllabus to
TODAY FOR A TURNKEY
[email protected] and you’ll
be entered to win an iPad Mini.
CLASSROOM SOLUTION OR MODIFY
If you don’t win, don’t worry – we’ll be
holding a new contest each semester.
IT TO SUIT YOUR TEACHING
APPROACH. FREE ONLINE AND
LOW-COST IN PRINT, OPENSTAX
Table of Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter1: Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1 Approximating Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 The Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.3 The Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
1.4 Integration Formulas and the Net Change Theorem . . . . . . . . . . . . . . . . . . . . . . 64
1.5 Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
1.6 Integrals Involving Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . . . 93
1.7 Integrals Resulting in Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . 106
Chapter2: Applications of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
2.1 Areas between Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
2.2 Determining Volumes by Slicing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
2.3 Volumes of Revolution: Cylindrical Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
2.4 Arc Length of a Curve and Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
2.5 Physical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
2.6 Moments and Centers of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
2.7 Integrals, Exponential Functions, and Logarithms . . . . . . . . . . . . . . . . . . . . . . . 219
2.8 Exponential Growth and Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
2.9 Calculus of the Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
Chapter3: Techniques of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
3.1 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
3.2 Trigonometric Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
3.3 Trigonometric Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
3.4 Partial Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
3.5 Other Strategies for Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
3.6 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
3.7 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
Chapter4: Introduction to Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 351
4.1 Basics of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
4.2 Direction Fields and Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
4.3 Separable Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
4.4 The Logistic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
4.5 First-order Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408
Chapter5: Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
5.1 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
5.2 Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
5.3 The Divergence and Integral Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471
5.4 Comparison Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485
5.5 Alternating Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496
5.6 Ratio and Root Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509
Chapter6: Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531
6.1 Power Series and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532
6.2 Properties of Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544
6.3 Taylor and Maclaurin Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561
6.4 Working with Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581
Chapter7: Parametric Equations and Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . 605
7.1 Parametric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606
7.2 Calculus of Parametric Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625
7.3 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642
7.4 Area and Arc Length in Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 662
7.5 Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671
Appendix A:Table of Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699
Appendix B:Table of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705
Appendix C:Review of Pre-Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819
This OpenStax book is available for free at http://cnx.org/content/col11965/1.2
Preface 1
PREFACE
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
Customization
Calculus Volume 2 is licensed under a Creative Commons Attribution Non-Commercial ShareAlike (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.
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.
Faculty also have the option of creating a customized version of their OpenStax book through the aerSelect platform.
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 openstax.org for a link to your book on aerSelect.
Errata
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 openstax.org. 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 openstax.org.
Format
You can access this textbook for free in web view or PDF through openstax.org, 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
2 Preface
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
ThroughoutCalculusVolume2youwillfindexamplesandexercisesthatpresentclassicalideasandtechniquesaswellas
modern applications and methods. Derivations and explanations are based on years of classroom experience on the part
oflong-timecalculusprofessors,strivingforabalanceofclarityandrigorthathasprovensuccessfulwiththeirstudents.
Motivationalapplicationscoverimportanttopicsinprobability,biology,ecology,business,andeconomics,aswellasareas
ofphysics,chemistry,engineering,andcomputerscience.StudentProjectsineachchaptergivestudentsopportunitiesto
exploreinterestingsidelightsinpureandappliedmathematics,fromshowingthatthenumbereisirrational,tocalculating
the center of mass of the Grand Canyon Skywalk or the terminal speed of a skydiver. Chapter Opening Applications
poseproblemsthataresolvedlaterinthechapter,usingtheideascoveredinthatchapter.Problemsincludethehydraulic
force against the Hoover Dam, and the comparison of the relative intensity of two earthquakes. Definitions, Rules, and
Theoremsare highlighted throughout the text, including over 60Proofsof theorems.
Assessments That Reinforce Key Concepts
In-chapterExampleswalkstudentsthroughproblemsbyposingaquestion,steppingoutasolution,andthenaskingstudents
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
indicatetheyaresuitableforsolutionbytechnology,includingcalculatorsorComputerAlgebraSystems(CAS).Answers
forselectedexercisesareavailableintheAnswerKeyatthebackofthebook.Thebookalsoincludesassessmentsatthe
end of each chapter so students can apply what they’ve learned through practice problems.
Early or Late Transcendentals
Calculus Volume 2 is designed to accommodate both Early and Late Transcendental approaches to calculus. Exponential
andlogarithmicfunctionsarepresentedinChapter2.IntegrationofthesefunctionsiscoveredinChapters1forinstructors
whowanttoinclude them withothertypesoffunctions. These discussions, however,areinseparate sections that canbe
skippedforinstructorswhoprefertowaituntiltheintegraldefinitionsaregivenbeforeteachingthecalculusderivationsof
exponentials and logarithms.
Comprehensive Art Program
Our art program is designed to enhance students’ understanding of concepts through clear and effective illustrations,
This OpenStax book is available for free at http://cnx.org/content/col11965/1.2