Calculus II { Tunc Geveci Copyright © 2011 by Tunc Geveci. All rights reserved. No part of this publication may be reprinted, reproduced, transmitted, or utilized in any form or by any electronic, mechanical, or other means, now known or hereaft er invented, including photocopying, microfi lming, and recording, or in any information retrieval system without the written permission of University Readers, Inc. First published in the United States of America in 2011 by Cognella, a division of University Readers, Inc. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identifi cation and explanation without intent to infringe. 15 14 13 12 11 1 2 3 4 5 Printed in the United States of America ISBN: 978-1-935551-44-7 Contents 6 Techniques of Integration 1 6.1 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 6.2 Integrals of Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 6.3 Integrals of Some Trigonometric and Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 6.4 Trigonometric and Hyperbolic Substitutions . . . . . . . . . . . . . . . . . . . . . 43 6.5 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 6.6 Improper Integrals: Part 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.7 Improper Integrals: Part 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 7 Applications of Integration 89 7.1 Volumes by Slices or Cylindrical Shells . . . . . . . . . . . . . . . . . . . . . . . . 89 7.2 Length and Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 7.3 Some Physical Applications of the Integral . . . . . . . . . . . . . . . . . . . . . . 111 7.4 The Integral and Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 8 Differential Equations 133 8.1 First-Order Linear Differential Equations . . . . . . . . . . . . . . . . . . . . . . 133 8.2 Applications of First-Order Linear Differential Equations . . . . . . . . . . . . . 148 8.3 Separable Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 8.4 Applications of Separable Differential Equations . . . . . . . . . . . . . . . . . . 171 8.5 Approximate Solutions and Slope Fields . . . . . . . . . . . . . . . . . . . . . . . 179 9 Infinite Series 187 9.1 Taylor Polynomials: Part 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 9.2 Taylor Polynomials: Part 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 9.3 The Concept of an Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 9.4 The Ratio Test and the Root Test . . . . . . . . . . . . . . . . . . . . . . . . . . 217 9.5 Power Series: Part 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 9.6 Power Series: Part 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 9.7 The Integral Test and Comparison Tests . . . . . . . . . . . . . . . . . . . . . . . 253 9.8 Conditional Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 9.9 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 10 Parametrized Curves and Polar Coordinates 285 10.1 Parametrized Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 10.2 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 10.3 Tangents and Area in Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . 305 10.4 Arc Length of Parametrized Curves. . . . . . . . . . . . . . . . . . . . . . . . . . 310 10.5 Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 10.6 Conic Sections in Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 325 iii iv CONTENTS H Taylor’s Formula for the Remainder 335 I Answers to Some Problems 341 J Basic Differentiation and Integration formulas 371 Preface This is the second volume of my calculus series, Calculus I, Calculus II and Calculus III. Thisseriesisdesignedfortheusualthreesemestercalculussequencethatthemajorityofscience andengineeringmajorsintheUnitedStatesarerequiredtotake. Somemajorsmayberequired to take only the first two parts of the sequence. Calculus I covers the usual topics of the first semester: Limits, continuity, the deriv- ative, the integral and special functions such exponential functions, logarithms, and inverse trigonometric functions. Calculus II covers the material of the second semester: Further techniques and applications of the integral, improper integrals, linear and separable first-order differential equations, infinite series, parametrized curves and polar coordinates. Calculus III covers topics in multivariable calculus: Vectors, vector-valued functions, directional derivatives, local linear approxima- tions,multipleintegrals,lineintegrals,surfaceintegrals,andthetheoremsofGreen, Gauss and Stokes. An important feature of my book is its focus on the fundamental concepts, essential functions and formulas of calculus. Students should not lose sight of the basic concepts and tools of calculus by being bombarded with functions and differentiation or antidifferentia- tion formulas that are not significant. I have written the examples and designed the exercises accordingly. I believe that "less is more". That approach enables one to demonstrate to the studentsthebeautyandutilityofcalculus,withoutclutteringitwithuglyexpressions. Another importantfeatureofmybookis the use of visualization as an integral part of the expo- sition. I believe that the most significant contribution of technology to the teaching of a basic coursesuchascalculushasbeentheeffortlessproductionofgraphicsofgoodquality. Numerical experiments are also helpful in explaining the basic ideas of calculus, and I have included such data. Remarks on some icons: Ihaveindicatedthe endof aproofby¥, theend of an exampleby ¤ and the end of a remark by ♦. Supplements: An instructors’ solution manual thatcontains thesolutions ofalltheprob- lemsisavailableasaPDFfilethatcanbesenttoaninstructorwhohasadoptedthebook. The studentwhopurchasesthebookcanaccessthestudents’ solutions manual thatcontainsthe solutions of odd numbered problems via www.cognella.com. Acknowledgments: ScientificWorkPlaceenabledmetotypethetextandthemathematical formulas easily in a seamless manner. Adobe Acrobat Pro has enabled me to convert the LaTeX files to pdf files. Mathematica has enabled me to import high quality graphics to my documents. I am grateful to the producers and marketers of such software without which I would not have had the patience to write and rewrite the material in these volumes. I would also like to acknowledge my gratitude to two wonderful mathematicians who have influenced me most by demonstrating the beauty of Mathematics and teaching me to write clearly and precisely: Errett Bishop and Stefan Warschawski. v vi PREFACE Last,butnottheleast,IamgratefultoSimlaforherencouragementandpatiencewhileIspent hours in front a computer screen. Tunc Geveci ([email protected]) San Diego, August 2010 Chapter 6 Techniques of Integration In this chapter we introduce an important technique of integration that is referred to as in- tegration by parts. We will focus on the integration of rational functions via partial fraction decompositions, the integration of various trigonometric and hyperbolic func- tions, and certain substitutions that are helpful in the integration of some expressions that involvethesquare-root. Wewilldiscussbasic approximation schemes forintegrals. Wewill alsodiscussthemeaningoftheso-calledimproper integralsthatinvolveunboundedintervals and/or functions with discontinuities. 6.1 Integration by Parts Integration by parts is the rule for indefinite and definite integrals that corresponds to the productrulefordifferentiation,justasthesubstitutionruleisthecounterpartofthechainrule. Theruleishelpfulintheevaluationofcertainintegralsandleadstousefulgeneralrelationships involving derivatives and integrals. Integration by Parts for Indefinite Integrals Assume that f and g are differentiable in the interval J. By the product rule, d df dg (f(x)g(x))= g(x)+f(x) dx dx dx for each x J. This is equivalent to the statement that f g+fg is an antiderivative of fg. 0 0 ∈ Thus, df dg f(x)g(x)= g(x)+f(x) dx dx dx Z µ ¶ for each x J. By the linearity of indefinite integrals, ∈ df dg f(x)g(x)= g(x)dx+ f(x) dx dx dx Z Z Therefore, dg df f(x) dx=f(x)g(x) g(x) dx dx − dx Z Z for each x J. This is the indefinite integral version of integration by parts: ∈ 1 2 CHAPTER 6. TECHNIQUES OF INTEGRATION INTEGRATION BY PARTS FOR DEFINITE INTEGRALS Assume that f and g are differentiable in the interval J. Then, dg df f(x) dx=f(x)g(x) g(x) dx dx − dx Z Z for each x J. ∈ We can use the ‘prime notation”, of course: f(x)g (x)dx=f(x)g(x) g(x)f (x)dx 0 0 − Z Z Example 1 a) Determine xe xdx − Z b) Check that your response to part a) is valid by differentiation. Solution a)Wewillsetf(x)=xanddg/dx=e x, andapplyintegration byparts, as statedin Theorem − 1. We have df d = (x)=1, dx dx and dg =e x g(x)= e xdx − − dx ⇔ Z The determination of g(x) is itself an antidifferentiation problem. We set u = x, so that − du/dx= 1. By the substitution rule, − du e xdx= e x( 1)dx= eu dx= eudu= eu+C = e x+C, − − − − − − dx − − − Z Z Z Z where C is an arbitrary constant. In the implementation of integration by parts, any antideriv- ative will do. Let us set g(x)= e x. Therefore, − − xe xdx= f(x)g (x)dx=f(x)g(x) f (x)g(x)dx − 0 0 − Z Z Z =x e x (1) e x dx − − − − − Z ¡ ¢ ¡ ¢ = xe x+ e xdx= xe x e x+C, − − − − − − − Z where C is an arbitrary constant. b) The expression xe xdx= xe x e x+C − − − − − Z is valid on the entire number line. Indeed, by the linearity of differentiation and the product rule, 6.1. INTEGRATION BY PARTS 3 d d d d xe x e x+C = xe x e x+ (C) − − − − dx − − −dx − dx dx ¡ ¢ d¡ ¢ d = (x) e x x e x +e x − − − − dx − dx µ ¶ µ ¶ = e x+xe x+e x =xe x − − − − − for each x R. The use of the product rule is not surprising, since we derived the formula for ∈ integration by parts from the product rule. ¤ The symbolic expression du du= dx dx is helpful in the implementation of the substitution rule. This symbolism is also helpful in the implementation of integration by parts. In the expression dg df f(x) dx=f(x)g(x) g(x) dx, dx − dx Z Z let us replace f(x) by u and g(x) by v. Thus, dv du u dx=uv v dx. dx − dx Z Z Let us also replace du dx dx by du, and dv dx dx by dv. Therefore, we can express the formula for integration by parts as follows: udv =uv vdu. − Z Z Note that dv v = dx= dv. dx Z Z Example 2 Determine xsin(4x)dx. Z Solution We will apply the formula for integration by parts in the form udv =uv vdu, − Z Z with u=x and dv =sin(4x)dx. Therefore, du du= dx=dx, dx