WIXLIB

Calculus IIFORDUMmIES‰by Mark Zegarelli

Calculus II For Dummies Published byWiley Publishing, Inc.111 River St.Hoboken, NJ 07030-5774www.wiley.comCopyright 2008 by Wiley Publishing, Inc., Indianapolis, IndianaPublished simultaneously in CanadaNo part of this publication may be reproduced, stored in a retrieval system, or transmitted in any formor by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior writtenpermission of the Publisher, or authorization through payment of the appropriate per-copy fee to theCopyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400, fax 978-646-8600.Requests to the Publisher for permission should be addressed to the Legal Department, Wiley Publishing,Inc., 10475 Crosspoint Blvd., Indianapolis, IN 46256, 317-572-3447, fax 317-572-4355, or online athttp://www.wiley.com/go/permissions.Trademarks: Wiley, the Wiley Publishing logo, For Dummies, the Dummies Man logo, A Reference for theRest of Us!, The Dummies Way, Dummies Daily, The Fun and Easy Way, Dummies.com and related tradedress are trademarks or registered trademarks of John Wiley & Sons, Inc. and/or its affiliates in the UnitedStates and other countries, and may not be used without written permission. All other trademarks are theproperty of their respective owners. Wiley Publishing, Inc., is not associated with any product or vendormentioned in this book.LIMIT OF LIABILITY/DISCLAIMER OF WARRANTY: THE PUBLISHER AND THE AUTHOR MAKE NO REPRESENTATIONS OR WARRANTIES WITH RESPECT TO THE ACCURACY OR COMPLETENESS OF THE CONTENTS OF THIS WORK AND SPECIFICALLY DISCLAIM ALL WARRANTIES, INCLUDING WITHOUTLIMITATION WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE. NO WARRANTY MAY BE CREATED OR EXTENDED BY SALES OR PROMOTIONAL MATERIALS. THE ADVICE AND STRATEGIES CONTAINED HEREIN MAY NOT BE SUITABLE FOR EVERY SITUATION. THIS WORK IS SOLD WITH THEUNDERSTANDING THAT THE PUBLISHER IS NOT ENGAGED IN RENDERING LEGAL, ACCOUNTING, OROTHER PROFESSIONAL SERVICES. IF PROFESSIONAL ASSISTANCE IS REQUIRED, THE SERVICES OF ACOMPETENT PROFESSIONAL PERSON SHOULD BE SOUGHT. NEITHER THE PUBLISHER NOR THEAUTHOR SHALL BE LIABLE FOR DAMAGES ARISING HEREFROM. THE FACT THAT AN ORGANIZATIONOR WEBSITE IS REFERRED TO IN THIS WORK AS A CITATION AND/OR A POTENTIAL SOURCE OF FURTHER INFORMATION DOES NOT MEAN THAT THE AUTHOR OR THE PUBLISHER ENDORSES THE INFORMATION THE ORGANIZATION OR WEBSITE MAY PROVIDE OR RECOMMENDATIONS IT MAY MAKE.FURTHER, READERS SHOULD BE AWARE THAT INTERNET WEBSITES LISTED IN THIS WORK MAY HAVECHANGED OR DISAPPEARED BETWEEN WHEN THIS WORK WAS WRITTEN AND WHEN IT IS READ.For general information on our other products and services, please contact our Customer CareDepartment within the U.S. at 800-762-2974, outside the U.S. at 317-572-3993, or fax 317-572-4002.For technical support, please visit www.wiley.com/techsupport.Wiley also publishes its books in a variety of electronic formats. Some content that appears in print maynot be available in electronic books.Library of Congress Control Number: 2008925786ISBN: 978-0-470-22522-6Manufactured in the United States of America10 9 8 7 6 5 4 3 2 1

About the AuthorMark Zegarelli is the author of Logic For Dummies (Wiley), Basic Math &Pre-Algebra For Dummies (Wiley), and numerous books of puzzles. He holdsdegrees in both English and math from Rutgers University, and lives in LongBranch, New Jersey, and San Francisco, California.DedicationFor my brilliant and beautiful sister, Tami. You are an inspiration.Author’s AcknowledgmentsMany thanks for the editorial guidance and wisdom of Lindsay Lefevere,Stephen Clark, and Sarah Faulkner of Wiley Publishing. Thanks also to theTechnical Editor, Jeffrey A. Oaks, PhD. Thanks especially to my friend DavidNacin, PhD, for his shrewd guidance and technical assistance.Much love and thanks to my family: Dr. Anthony and Christine Zegarelli,Mary Lou and Alan Cary, Joe and Jasmine Cianflone, and Deseret MoctezumaRackham and Janet Rackham. Thanksgiving is at my place this year!And, as always, thank you to my partner, Mark Dembrowski, for your constant wisdom, support, and love.

Publisher’s AcknowledgmentsWe’re proud of this book; please send us your comments through our Dummies online registrationform located at www.dummies.com/register/.Some of the people who helped bring this book to market include the following:Acquisitions, Editorial, and MediaDevelopmentComposition ServicesProject Coordinator: Katie KeyProject Editor: Stephen R. ClarkLayout and Graphics: Carrie A. CesaviceAcquisitions Editor: Lindsay Sandman LefevereProofreaders: Laura Albert, Laura L. BowmanSenior Copy Editor: Sarah FaulknerIndexer: Broccoli Information ManagementEditorial Program Coordinator:Erin Calligan MooneySpecial HelpTechnical Editor: Jeffrey A. Oaks, PhDDavid Nacin, PhDEditorial Manager: Christine Meloy BeckEditorial Assistants: Joe Niesen, David LuttonCover Photos: ComstockCartoons: Rich Tennant(www.the5thwave.com)Publishing and Editorial for Consumer DummiesDiane Graves Steele, Vice President and Publisher, Consumer DummiesJoyce Pepple, Acquisitions Director, Consumer DummiesKristin A. Cocks, Product Development Director, Consumer DummiesMichael Spring, Vice President and Publisher, TravelKelly Regan, Editorial Director, TravelPublishing for Technology DummiesAndy Cummings, Vice President and Publisher, Dummies Technology/General UserComposition ServicesGerry Fahey, Vice President of Production ServicesDebbie Stailey, Director of Composition Services

Contents at a GlanceIntroduction .1Part I: Introduction to Integration .9Chapter 1: An Aerial View of the Area Problem .11Chapter 2: Dispelling Ghosts from the Past:A Review of Pre-Calculus and Calculus I .37Chapter 3: From Definite to Indefinite: The Indefinite Integral .73Part II: Indefinite Integrals .103Chapter 4: Instant Integration: Just Add Water (And C) .105Chapter 5: Making a Fast Switch: Variable Substitution .117Chapter 6: Integration by Parts .135Chapter 7: Trig Substitution: Knowing All the (Tri)Angles .151Chapter 8: When All Else Fails: Integration with Partial Fractions .173Part III: Intermediate Integration Topics .195Chapter 9: Forging into New Areas: Solving Area Problems .197Chapter 10: Pump up the Volume: Using Calculus to Solve 3-D Problems .219Part IV: Infinite Series .241Chapter 11: Following a Sequence, Winning the Series .243Chapter 12: Where Is This Going? Testing for Convergence and Divergence .261Chapter 13: Dressing up Functions with the Taylor Series .283Part V: Advanced Topics .305Chapter 14: Multivariable Calculus .307Chapter 15: What’s So Different about Differential Equations? .327Part VI: The Part of Tens .341Chapter 16: Ten “Aha!” Insights in Calculus II .343Chapter 17: Ten Tips to Take to the Test .349Index .353

Table of ContentsIntroduction.1About This Book .1Conventions Used in This Book .3What You’re Not to Read .3Foolish Assumptions .3How This Book Is Organized .4Part I: Introduction to Integration .4Part II: Indefinite Integrals .4Part III: Intermediate Integration Topics .5Part IV: Infinite Series .5Part V: Advanced Topics .6Part VI: The Part of Tens .7Icons Used in This Book .7Where to Go from Here .8Part I: Introduction to Integration .9Chapter 1: An Aerial View of the Area Problem . . . . . . . . . . . . . . . . . .11Checking out the Area .12Comparing classical and analytic geometry .12Discovering a new area of study .13Generalizing the area problem .15Finding definite answers with the definite integral .16Slicing Things Up .19Untangling a hairy problem by using rectangles .20Building a formula for finding area .22Defining the Indefinite .27Solving Problems with Integration .28We can work it out: Finding the area between curves .29Walking the long and winding road .29You say you want a revolution .30Understanding Infinite Series .31Distinguishing sequences and series .31Evaluating series .32Identifying convergent and divergent series .32Advancing Forward into Advanced Math .33Multivariable calculus .33Differential equations .34Fourier analysis .34Numerical analysis .34

viiiCalculus II For DummiesChapter 2: Dispelling Ghosts from the Past:A Review of Pre-Calculus and Calculus I . . . . . . . . . . . . . . . . . . . . . . .37Forgotten but Not Gone: A Review of Pre-Calculus .38Knowing the facts on factorials .38Polishing off polynomials .39Powering through powers (exponents) .39Noting trig notation .41Figuring the angles with radians .42Graphing common functions .43Asymptotes .47Transforming continuous functions .47Identifying some important trig identities .48Polar coordinates .50Summing up sigma notation .51Recent Memories: A Review of Calculus I .53Knowing your limits .53Hitting the slopes with derivatives .55Referring to the limit formula for derivatives .56Knowing two notations for derivatives .56Understanding differentiation .57Finding Limits by Using L’Hospital’s Rule .64Understanding determinate and indeterminate forms of limits .65Introducing L’Hospital’s Rule .66Alternative indeterminate forms .68Chapter 3: From Definite to Indefinite: The Indefinite Integral . . . . .73Approximate Integration .74Three ways to approximate area with rectangles .74The slack factor .78Two more ways to approximate area .79Knowing Sum-Thing about Summation Formulas .83The summation formula for counting numbers .83The summation formula for square numbers .84The summation formula for cubic numbers .84As Bad as It Gets: Calculating Definite Integrals byUsing the Riemann Sum Formula .85Plugging in the limits of integration .86Expressing the function as a sum in terms of i and n .86Calculating the sum .88Solving the problem with a summation formula .88Evaluating the limit .89Light at the End of the Tunnel: The FundamentalTheorem of Calculus .89

Table of ContentsUnderstanding the Fundamental Theorem of Calculus .91What’s slope got to do with it? .92Introducing the area function .92Connecting slope and area mathematically .94Seeing a dark side of the FTC .95Your New Best Friend: The Indefinite Integral .95Introducing anti-differentiation .96Solving area problems without the Riemann sum formula .97Understanding signed area .99Distinguishing definite and indefinite integrals .101Part II: Indefinite Integrals .103Chapter 4: Instant Integration: Just Add Water (And C) . . . . . . . . . .105Evaluating Basic Integrals .106Using the 17 basic anti-derivatives for integrating .106Three important integration rules .107What happened to the other rules? .110Evaluating More Difficult Integrals .110Integrating polynomials .110Integrating rational expressions .111Using identities to integrate trig functions .112Understanding Integrability .113Understanding two red herrings of integrability .114Understanding what integrable really means .115Chapter 5: Making a Fast Switch: Variable Substitution . . . . . . . . .117Knowing How to Use Variable Substitution .118Finding the integral of nested functions .118Finding the integral of a product .120Integrating a function multipliedby a set of nested functions .121Recognizing When to Use Substitution .123Integrating nested functions .123Knowing a shortcut for nested functions .125Substitution when one part of a functiondifferentiates to the other part .129Using Substitution to Evaluate Definite Integrals .132Chapter 6: Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .135Introducing Integration by Parts .135Reversing the Product Rule .136Knowing how to integrate by parts .137Knowing when to integrate by parts .138ix