Single Variable Calculus: Vol. 2, Early Transcendentals

553 Pages·2010·18.262 MB·English

by  James Stewart

97909_FrontEP_FrontEP_pF2_RefPage1-2_97909_FrontEP_FrontEP_pF2_RefPage1-2 9/24/10 5:36 PM Page 1
REFERENCE PAGE 1
ALGEBRA GEOMETRY
e Arithmetic Operations Geometric Formulas
c
n
ere a(cid:2)b(cid:4)c(cid:3)(cid:2)ab(cid:4)ac a (cid:4) c (cid:2) ad(cid:4)bc Formulas for area A, circumference C, and volume V:
ef b d bd
ep for r a(cid:4)c (cid:2) a (cid:4) c ab (cid:2) a (cid:2) d (cid:2) ad TAri(cid:2)an12gbleh CAir(cid:2)cle(cid:3)r2 SAe(cid:2)cto12rr o2f(cid:6) Circle
e b b b c b c bc (cid:2)1absin (cid:6) C(cid:2)2(cid:3)r s(cid:2)r(cid:6) (cid:2)(cid:6)in radians(cid:3)
k 2
d d
n
a
ere Exponents and Radicals a h r s
h
ut xm ¨ r
C xmxn(cid:2)xm(cid:4)n (cid:2)xm(cid:5)n b ¨
xn r
1
(cid:2)xm(cid:3)n(cid:2)xmn x(cid:5)n(cid:2)
xn
(cid:4) (cid:5)
x n xn Sphere Cylinder Cone
(cid:2)xy(cid:3)n(cid:2)xnyn (cid:2)
y yn V(cid:2)4(cid:3)r3 V(cid:2)(cid:3)r2h V(cid:2)1(cid:3)r2h
3 3
x1(cid:8)n(cid:2)snx xm(cid:8)n(cid:2)snxm(cid:2)(snx)m A(cid:2)4(cid:3)r2 A(cid:2)(cid:3)rsr2(cid:4)h2
(cid:7)
snxy(cid:2)snxsny n x (cid:2) snx r
y sny
r h
h
Factoring Special Polynomials
r
x2(cid:5)y2(cid:2)(cid:2)x(cid:4)y(cid:3)(cid:2)x(cid:5)y(cid:3)
x3(cid:4)y3(cid:2)(cid:2)x(cid:4)y(cid:3)(cid:2)x2(cid:5)xy(cid:4)y2(cid:3)
x3(cid:5)y3(cid:2)(cid:2)x(cid:5)y(cid:3)(cid:2)x2(cid:4)xy(cid:4)y2(cid:3)
Distance and Midpoint Formulas
Binomial Theorem
Distance between P(cid:2)x, y(cid:3)and P(cid:2)x, y(cid:3):
1 1 1 2 2 2
(cid:2)x(cid:4)y(cid:3)2(cid:2)x2(cid:4)2xy(cid:4)y2 (cid:2)x(cid:5)y(cid:3)2(cid:2)x2(cid:5)2xy(cid:4)y2
(cid:2)x(cid:4)y(cid:3)3(cid:2)x3(cid:4)3x2y(cid:4)3xy2(cid:4)y3 d(cid:2)s(cid:2)x2(cid:5)x1(cid:3)2(cid:4)(cid:2)y2(cid:5)y1(cid:3)2
(cid:2)x(cid:5)y(cid:3)3(cid:2)x3(cid:5)3x2y(cid:4)3xy2(cid:5)y3 (cid:4) (cid:5)
(cid:2)x(cid:4)y(cid:3)n(cid:2)xn(cid:4)nxn(cid:5)1y(cid:4) n(cid:2)n(cid:5)1(cid:3) xn(cid:5)2y2 Midpoint of P1P2: x1(cid:4)2 x2, y1(cid:4)2 y2
2
(cid:4) (cid:5)
n
(cid:4)(cid:10)(cid:10)(cid:10)(cid:4) xn(cid:5)kyk(cid:4)(cid:10)(cid:10)(cid:10)(cid:4)nxyn(cid:5)1(cid:4)yn
k
(cid:4) (cid:5) Lines
n n(cid:2)n(cid:5)1(cid:3)(cid:10)(cid:10)(cid:10)(cid:2)n (cid:5)k(cid:4)1(cid:3)
where (cid:2) Slope of line through P(cid:2)x, y(cid:3)and P(cid:2)x, y(cid:3):
k 1(cid:2)2(cid:2)3(cid:2)(cid:10)(cid:10)(cid:10)(cid:2)k 1 1 1 2 2 2
Quadratic Formula m(cid:2) y2(cid:5)y1
x (cid:5)x
2 1
(cid:5)b(cid:9)sb2(cid:5)4ac
If ax2(cid:4)bx(cid:4)c(cid:2)0, then x(cid:2) .
2a Point-slope equation of line through P(cid:2)x, y(cid:3)with slope m:
1 1 1
Inequalities and Absolute Value y(cid:5)y (cid:2)m(cid:2)x(cid:5)x(cid:3)
1 1
If a(cid:7)band b(cid:7)c, then a(cid:7)c.
Slope-intercept equation of line with slope mand y-intercept b:
If a(cid:7)b, then a(cid:4)c(cid:7)b(cid:4)c.
If a(cid:7)band c(cid:8)0, then ca(cid:7)cb. y(cid:2)mx(cid:4)b
If a(cid:7)band c(cid:7)0, then ca(cid:8)cb.
If a(cid:8)0, then Circles
(cid:6)x(cid:6)(cid:2)a means x(cid:2)a or x(cid:2)(cid:5)a
(cid:6)x(cid:6)(cid:7)a means (cid:5)a(cid:7)x(cid:7)a Equation of the circle with center (cid:2)h, k(cid:3)and radius r:
(cid:6)x(cid:6)(cid:8)a means x(cid:8)a or x(cid:7)(cid:5)a (cid:2)x(cid:5)h(cid:3)2(cid:4)(cid:2)y(cid:5)k(cid:3)2(cid:2)r2
97909_FrontEP_FrontEP_pF2_RefPage1-2_97909_FrontEP_FrontEP_pF2_RefPage1-2 9/24/10 5:36 PM Page 2
REFERENCE PAGE 2
TRIGONOMETRY
Angle Measurement Fundamental Identities
(cid:2)radians(cid:2)180(cid:3) 1 1
s csc (cid:4)(cid:2) sec (cid:4)(cid:2)
r sin (cid:4) cos (cid:4)
(cid:2) 180(cid:3)
1(cid:3)(cid:2) rad 1rad(cid:2) ¨
180 (cid:2) sin (cid:4) cos (cid:4)
s(cid:2)r(cid:4) r tan (cid:4)(cid:2) cos (cid:4) cot (cid:4)(cid:2) sin (cid:4)
(cid:3)(cid:4)in radians(cid:4) cot (cid:4)(cid:2) 1 sin2(cid:4)(cid:5)cos2(cid:4)(cid:2)1
tan (cid:4)
Right Angle Trigonometry
1(cid:5)tan2(cid:4)(cid:2)sec2(cid:4) 1(cid:5)cot2(cid:4)(cid:2)csc2(cid:4)
opp hyp
sin (cid:4)(cid:2) hyp csc (cid:4)(cid:2) opp hyp sin(cid:3)(cid:6)(cid:4)(cid:4)(cid:2)(cid:6)sin (cid:4) cos(cid:3)(cid:6)(cid:4)(cid:4)(cid:2)cos (cid:4)
opp (cid:5) (cid:6)
adj hyp (cid:2)
cos (cid:4)(cid:2) sec (cid:4)(cid:2) ¨ tan(cid:3)(cid:6)(cid:4)(cid:4)(cid:2)(cid:6)tan (cid:4) sin (cid:6)(cid:4) (cid:2)cos (cid:4)
hyp adj 2
adj (cid:5) (cid:6) (cid:5) (cid:6)
opp adj
tan (cid:4)(cid:2) cot (cid:4)(cid:2) (cid:2) (cid:2)
adj opp cos (cid:6)(cid:4) (cid:2)sin (cid:4) tan (cid:6)(cid:4) (cid:2)cot (cid:4)
2 2
Trigonometric Functions
The Law of Sines B
sin (cid:4)(cid:2) y csc (cid:4)(cid:2) r y
r y sin A (cid:2) sin B (cid:2) sin C a
x r (x, y) a b c
cos (cid:4)(cid:2) sec (cid:4)(cid:2) r C
r x c
y x ¨ The Law of Cosines
tan (cid:4)(cid:2) x cot (cid:4)(cid:2) y x b
a2(cid:2)b2(cid:5)c2(cid:6)2bccos A
Graphs of Trigonometric Functions b2(cid:2)a2(cid:5)c2(cid:6)2accos B
y y y y=tan x c2(cid:2)a2(cid:5)b2(cid:6)2abcos C A
y=sin x y=cos x
1 1
Addition and Subtraction Formulas
π 2π 2π
x π 2πx π x sin(cid:3)x(cid:5)y(cid:4)(cid:2)sin x cos y(cid:5)cos x sin y
_1 _1 sin(cid:3)x(cid:6)y(cid:4)(cid:2)sin x cos y(cid:6)cos x sin y
cos(cid:3)x(cid:5)y(cid:4)(cid:2)cos x cos y(cid:6)sin x sin y
y y=csc x y y=sec x y y=cot x cos(cid:3)x(cid:6)y(cid:4)(cid:2)cos x cos y(cid:5)sin x sin y
tan x(cid:5)tan y
tan(cid:3)x(cid:5)y(cid:4)(cid:2)
1 1 1(cid:6)tan x tan y
tan x(cid:6)tan y
π 2πx π 2πx π 2πx tan(cid:3)x(cid:6)y(cid:4)(cid:2) 1(cid:5)tan x tan y
_1 _1
Double-Angle Formulas
sin 2x(cid:2)2 sin x cos x
Trigonometric Functions of Important Angles cos 2x(cid:2)cos2x(cid:6)sin2x(cid:2)2 cos2x(cid:6)1(cid:2)1(cid:6)2 sin2x
(cid:4) radians sin (cid:4) cos (cid:4) tan (cid:4) 2 tan x
tan 2x(cid:2)
0(cid:3) 0 0 1 0 1(cid:6)tan2x
30(cid:3) (cid:2)(cid:2)6 1(cid:2)2 s3(cid:2)2 s3(cid:2)3
45(cid:3) (cid:2)(cid:2)4 s2(cid:2)2 s2(cid:2)2 1 Half-Angle Formulas
60(cid:3) (cid:2)(cid:2)3 s3(cid:2)2 1(cid:2)2 s3 1(cid:6)cos 2x 1(cid:5)cos 2x
90(cid:3) (cid:2)(cid:2)2 1 0 — sin2x(cid:2) 2 cos2x(cid:2) 2
98706_FMSVETVol2_FMSVETVol2_pi-xx.qk_98706_FMSVETVol2_FMSVETVol2_pi-xx 10/15/10 1:35 PM Page i
S I N G L E V A R I A B L E
C A L C U L U S
E A R L Y T R A N S C E N D E N T A L S
SEVENTH EDITION
VOLUME 2: CHAPTERS 5–11
This page intentionally left blank
98706_FMSVETVol2_FMSVETVol2_pi-xx.qk_98706_FMSVETVol2_FMSVETVol2_pi-xx 10/15/10 1:35 PM Page iii
S I N G L E V A R I A B L E
C A L C U L U S
E A R L Y T R A N S C E N D E N T A L S
SEVENTH EDITION
VOLUME 2: CHAPTERS 5–11
JAMES STEWART
McMASTER UNIVERSITY
AND
UNIVERSITY OF TORONTO
Australia.Brazil.Japan.Korea.Mexico.Singapore.Spain.UnitedKingdom.United States
This is an electronic version of the print textbook. Due to electronic rights restrictions, some third party content may be suppressed.
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience.
The publisher reserves the right to remove content from this title at any time if subsequent rights restrictions require it.
For valuable information on pricing, previous editions, changes to current editions, and alternate formats,
please visit www.cengage.com/highered to search by ISBN#, author, title, or keyword for materials in your areas of interest.
98706_FMSVETVol2_FMSVETVol2_pi-xx.qk_98706_FMSVETVol2_FMSVETVol2_pi-xx 10/15/10 1:35 PM Page iv
Single Variable Calculus: Early Transcendentals,Volume 2 © 2012, 2008Brooks/Cole, Cengage Learning
Seventh Edition
ALL RIGHTS RESERVED. No part of this work covered by the copy-
James Stewart
right herein may be reproduced, transmitted, stored, or used in any
form or by any means graphic, electronic, or mechanical, including
Executive Editor: Liz Covello
but not limited to photocopying, recording, scanning, digitizing,
AssistantEditor: Liza Neustaetter taping, Web distribution, information networks, or information stor-
EditorialAssistant: Jennifer Staller age and retrieval systems, except as permitted under Section 107or
108of the 1976United States Copyright Act, without the prior writ-
Media Editor: Maureen Ross
ten permission of the publisher.
MarketingManager: Jennifer Jones
MarketingCoordinator: Michael Ledesma For product information and technology assistance, contact us at
MarketingCommunicationsManager: Mary Anne Payumo Cengage Learning Customer & Sales Support, 1-800-354-9706.
Content ProjectManager: Cheryll Linthicum For permission to use material from this text or product,
submit all requests online at www.cengage.com/permissions.
ArtDirector: Vernon T. Boes
PrintBuyer: Becky Cross Further permissions questions can be e-mailed to
[email protected].
Rights Acquisitions Specialist: Don Schlotman
ProductionService: TECH·arts
Library of Congress Control Number: 2010936600
TextDesigner: TECH·arts
PhotoResearcher: Terri Wright, www.terriwright.com ISBN-13: 978-0-538-49870-8
CopyEditor: Kathi Townes ISBN-10: 0-538-49870-6
CoverDesigner: Irene Morris
CoverIllustration: Irene Morris
Brooks/Cole
Compositor: Stephanie Kuhns, TECH·arts 20Davis Drive
Belmont, CA 94002-3098
USA
Cengage Learning is a leading provider of customized learning solu-
tions with office locations around the globe, including Singapore, the
United Kingdom, Australia, Mexico, Brazil, and Japan. Locate your
local office at www.cengage.com/global.
Cengage Learning products are represented in Canada by Nelson
Education, Ltd.
To learn more about Brooks/Cole, visit
www.cengage.com/brookscole.
Purchase any of our products at your local college store or at our
preferred online store www.cengagebrain.com.
Trademarks
ExamView®and ExamViewPro®are registered trademarks
of FSCreations, Inc.
Windows is a registered trademark of the Microsoft Corporation
and used herein under license.
Macintosh and Power Macintosh are registered trademarks of
Apple Computer, Inc. Used herein under license.
Derive is a registered trademark of Soft Warehouse, Inc.
Maple is a registered trademark of Waterloo Maple, Inc.
0
1
Mathematica is a registered trademark of Wolfram Research, Inc. T
0
Tools for Enriching is a trademark used herein under license. K1
Printed in the United States of America
1 2 3 4 5 6 7 14 13 12 11 10
98706_FMSVETVol2_FMSVETVol2_pi-xx.qk_98706_FMSVETVol2_FMSVETVol2_pi-xx 10/15/10 1:36 PM Page v
Contents
Preface xi
To the Student xxii
Diagnostic Tests xxiv
5 Integrals        359
5.1 Areas and Distances 360
5.2 The Definite Integral 371
Discovery Project Area Functions 385
N
5.3 The Fundamental Theorem of Calculus 386
5.4 Indefinite Integrals and the Net Change Theorem 397
Writing Project Newton, Leibniz, and the Invention of Calculus 406
N
5.5 The Substitution Rule 407
Review 415
Problems Plus 419
6 Applications of Integration        421
6.1 Areas Between Curves 422
Applied Project The Gini Index 429
N
6.2 Volumes 430
6.3 Volumes by Cylindrical Shells 441
6.4 Work 446
6.5 Average Value of a Function 451
Applied Project Calculus and Baseball 455
N
Applied Project Where to Sit at the Movies 456
N
Review 457
Problems Plus 459
v
98706_FMSVETVol2_FMSVETVol2_pi-xx.qk_98706_FMSVETVol2_FMSVETVol2_pi-xx 10/15/10 1:36 PM Page vi
vi CONTENTS
7 Techniques of Integration        463
7.1 Integration by Parts 464
7.2 Trigonometric Integrals 471
7.3 Trigonometric Substitution 478
7.4 Integration of Rational Functions by Partial Fractions 484
7.5 Strategy for Integration 494
7.6 Integration Using Tables and Computer Algebra Systems 500
Discovery Project Patterns in Integrals 505
N
7.7 Approximate Integration 506
7.8 Improper Integrals 519
Review 529
Problems Plus 533
8 Further Applications of Integration        537
8.1 Arc Length 538
Discovery Project Arc Length Contest 545
N
8.2 Area of a Surface of Revolution 545
Discovery Project Rotating on a Slant 551
N
8.3 Applications to Physics and Engineering 552
Discovery Project Complementary Coffee Cups 562
N
8.4 Applications to Economics and Biology 563
8.5 Probability 568
Review 575
Problems Plus 577