Calculus II For Dummies

An easy-to-understand primer on advanced calculus topics Calculus II is a prerequisite for many popular college majors, including pre-med, engineering, and physics. Calculus II For Dummies offers expert instruction, advice, and tips to help second semester calculus students get a handle on the subject and ace their exams. It covers intermediate calculus topics in plain English, featuring in-depth coverage of integration, including substitution, integration techniques and when to use them, approximate integration, and improper integrals. This hands-on guide also covers sequences and series, with introductions to multivariable calculus, differential equations, and numerical analysis. Best of all, it includes practical exercises designed to simplify and enhance understanding of this complex subject. Introduction to integration Indefinite integrals Intermediate Integration topics Infinite series Advanced topics Practice exercises Confounded by curves? Perplexed by polynomials? This plain-English guide to Calculus II will set you straight!

Table of Contents:
Introduction 1 About This Book 1 Conventions Used in This Book 3 What You’re Not to Read 3 Foolish Assumptions 3 How This Book Is Organized 4 Part I: Introduction to Integration 4 Part II: Indefinite Integrals 4 Part III: Intermediate Integration Topics 5 Part IV: Infinite Series 5 Part V: Advanced Topics 6 Part VI: The Part of Tens 7 Icons Used in This Book 7 Where to Go from Here 7 Part I: Introduction to Integration 9 Chapter 1: An Aerial View of the Area Problem 11 Checking Out the Area 12 Comparing classical and analytic geometry 12 Discovering a new area of study 13 Generalizing the area problem 15 Finding definite answers with the definite integral 16 Slicing Things Up 19 Untangling a hairy problem using rectangles 20 Building a formula for finding area 22 Defining the Indefinite 28 Solving Problems with Integration 29 We can work it out: Finding the area between curves 29 Walking the long and winding road 30 You say you want a revolution 31 Understanding Infinite Series 31 Distinguishing sequences and series 32 Evaluating series 32 Identifying convergent and divergent series 33 Advancing Forward into Advanced Math 34 Multivariable calculus 34 Differential equations 35 Fourier analysis 35 Numerical analysis 35 Chapter 2: Dispelling Ghosts from the Past: A Review of Pre-Calculus and Calculus I 37 Forgotten but Not Gone: A Review of Pre-Calculus 38 Knowing the facts on factorials 38 Polishing off polynomials 39 Powering through powers (exponents) 39 Noting trig notation 41 Figuring the angles with radians 42 Graphing common functions 43 Asymptotes 47 Transforming continuous functions 48 Identifying some important trig identities 48 Polar coordinates 50 Summing up sigma notation 51 Recent Memories: A Review of Calculus I 53 Knowing your limits 53 Hitting the slopes with derivatives 55 Referring to the limit formula for derivatives 56 Knowing two notations for derivatives 56 Understanding differentiation 57 Finding Limits Using L’Hopital’s Rule 65 Understanding determinate and indeterminate forms of limits 65 Introducing L’Hopital’s Rule 67 Alternative indeterminate forms 68 Chapter 3: From Definite  to Indefinite : The Indefinite  Integral 73 Approximate Integration 74 Three ways to approximate area with rectangles 74 The slack factor 78 Two more ways to approximate area 79 Knowing Sum-Thing about Summation Formulas 83 The summation formula for counting numbers 83 The summation formula for square numbers 84 The summation formula for cubic numbers 84 As Bad as It Gets: Calculating Definite Integrals Using the Riemann Sum Formula 85 Plugging in the limits of integration 86 Expressing the function as a sum in terms of i and n 86 Calculating the sum 88 Solving the problem with a summation formula 89 Evaluating the limit 89 Light at the End of the Tunnel: The Fundamental Theorem of Calculus 90 Understanding the Fundamental Theorem of Calculus 92 What’s slope got to do with it? 92 Introducing the area function 93 Connecting slope and area mathematically 95 Seeing a dark side of the FTC 95 Your New Best Friend: The Indefinite Integral 96 Introducing anti-differentiation 97 Solving area problems without the Riemann sum formula 98 Understanding signed area 100 Distinguishing definite and indefinite integrals 101 Part II: Indefinite Integrals 103 Chapter 4: Instant Integration: Just Add Water (And C) 105 Evaluating Basic Integrals 106 Using the 17 basic anti-derivatives for integrating 106 Three important integration rules 108 What happened to the other rules? 110 Evaluating More Diffi cult Integrals 110 Integrating polynomials 111 Integrating rational expressions 111 Using identities to integrate trig functions 112 Understanding Integrability 114 Taking a look at two red herrings of integrability 114 Getting an idea of what integrable really means 115 Chapter 5: Making a Fast Switch: Variable Substitution 117 Knowing How to Use Variable Substitution 117 Finding the integral of nested functions 118 Determining the integral of a product 120 Integrating a function multiplied by a set of nested functions 121 Recognizing When to Use Substitution 123 Integrating nested functions 123 Knowing a shortcut for nested functions 125 Substitution when one part of a function differentiates to the other part 129 Using Substitution to Evaluate Definite Integrals 132 Chapter 6: Integration by Parts 135 Introducing Integration by Parts 135 Reversing the Product Rule 136 Knowing how to integrate by parts 137 Knowing when to integrate by parts 138 Integrating by Parts with the DI-agonal Method 140 Looking at the DI-agonal chart 140 Using the DI-agonal method 140 Chapter 7: Trig Substitution: Knowing All the (Tri)Angles 151 Integrating the Six Trig Functions 151 Integrating Powers of Sines and Cosines 152 Odd powers of sines and cosines 152 Even powers of sines and cosines 154 Integrating Powers of Tangents and Secants 155 Even powers of secants with tangents 155 Odd powers of tangents with secants 156 Odd powers of tangents without secants 156 Even powers of tangents without secants 157 Even powers of secants without tangents 157 Odd powers of secants without tangents 157 Even powers of tangents with odd powers of secants 159 Integrating Powers of Cotangents and Cosecants 159 Integrating Weird Combinations of Trig Functions 160 Using Trig Substitution 162 Distinguishing three cases for trig substitution 163 Integrating the three cases 164 Knowing when to avoid trig substitution 171 Chapter 8: When All Else Fails: Integration with Partial Fractions 173 Strange but True: Understanding Partial Fractions 174 Looking at partial fractions 174 Using partial fractions with rational expressions 175 Solving Integrals by Using Partial Fractions 176 Setting up partial fractions case by case 177 Knowing the ABCs of fi nding unknowns 181 Integrating partial fractions 184 Integrating Improper Rationals 188 Distinguishing proper and improper rational expressions 188 Recalling polynomial division 189 Trying out an example 192 Part III: Intermediate Integration Topics 195 Chapter 9: Forging into New Areas: Solving Area Problems 197 Breaking Us in Two 198 Improper Integrals 199 Getting horizontal 199 Going vertical 202 Solving Area Problems with More Than One Function 204 Finding the area under more than one function 205 Finding the area between two functions 206 Looking for a sign 209 Measuring unsigned area between curves with a quick trick 211 The Mean Value Theorem for Integrals 213 Calculating Arc Length 215 Chapter 10: Pump Up the Volume: Using Calculus to Solve 3-D Problems 219 Slicing Your Way to Success 220 Finding the volume of a solid with congruent cross sections 220 Finding the volume of a solid with similar cross sections 221 Measuring the volume of a pyramid 222 Measuring the volume of a weird solid 224 Turning a Problem on Its Side 225 Two Revolutionary Problems 227 Solidifying your understanding of solids of revolution 227 Skimming the surface of revolution 229 Finding the Space Between 231 Playing the Shell Game 234 Peeling and measuring a can of soup 235 Using the shell method 237 Knowing When and How to Solve 3-D Problems 238 Part IV: Infinite Series 241 Chapter 11: Following a Sequence, Winning the Series 243 Introducing Infinite Sequences 244 Understanding notations for sequences 244 Looking at converging and diverging sequences 246 Introducing Infinite Series 247 Getting Comfy with Sigma Notation 249 Writing sigma notation in expanded form 250 Seeing more than one way to use sigma notation 250 Discovering the Constant Multiple Rule for series 251 Examining the Sum Rule for series 252 Connecting a Series with Its Two Related Sequences 252 A series and its defining sequence 253 A series and its sequences of partial sums 253 Recognizing Geometric Series and P-Series 255 Getting geometric series 255 Pinpointing p-series 258 Chapter 12: Where Is This Going? Testing for Convergence and Divergence 261 Starting at the Beginning 262 Using the nth-Term Test for Divergence 263 Let Me Count the Ways 263 One-way tests 263 Two-way tests 264 Choosing Comparison Tests 264 Getting direct answers with the direct comparison test 265 Testing your limits with the limit comparison test 268 Two-Way Tests for Convergence and Divergence 270 Integrating a solution with the integral test 271 Rationally solving problems with the ratio test 273 Rooting out answers with the root test 274 Looking at Alternating Series 275 Eyeballing two forms of the basic alternating series 276 Making new series from old ones 276 Alternating series based on convergent positive series 277 Checking out the alternating series test 278 Understanding absolute and conditional convergence 280 Testing alternating series 282 Chapter 13: Dressing Up Functions with the Taylor Series 283 Elementary Functions 284 Knowing two drawbacks of elementary functions 284 Appreciating why polynomials are so friendly 285 Representing elementary functions as polynomials 285 Representing elementary functions as series 285 Power Series: Polynomials on Steroids 286 Integrating power series 287 Understanding the interval of convergence 288 Expressing Functions as Series 291 Expressing sin x as a series 291 Expressing cos x as a series 293 Introducing the Maclaurin Series 294 Introducing the Taylor Series 297 Computing with the Taylor series 298 Examining convergent and divergent Taylor series 299 Expressing functions versus approximating functions 301 Calculating error bounds for Taylor polynomials 302 Understanding Why the Taylor Series Works 304 Part V: Advanced Topics 307 Chapter 14: Multivariable Calculus 309 Visualizing Vectors 310 Understanding vector basics 310 Distinguishing vectors and scalars 312 Calculating with vectors 312 Leaping to Another Dimension 316 Understanding 3-D Cartesian coordinates 316 Using alternative 3-D coordinate systems 318 Functions of Several Variables 321 Partial Derivatives 322 Measuring slope in three dimensions 323 Evaluating partial derivatives 323 Multiple Integrals 325 Measuring volume under a surface 325 Evaluating multiple integrals 326 Chapter 15: What’s So Different about Differential Equations? 329 Basics of Differential Equations 330 Classifying DEs 330 Looking more closely at DEs 333 Solving Differential Equations 336 Solving separable equations 336 Solving initial-value problems (IVPs) 337 Using an integrating factor 339 Part VI: The Part of Tens 343 Chapter 16: Ten “Aha!” Insights in Calculus II 345 Integrating Means Finding the Area 345 When You Integrate, Area Means Signed Area 346 Integrating Is Just Fancy Addition 346 Integration Uses Infinitely Many Infinitely Thin Slices 346 Integration Contains a Slack Factor 347 A Definite Integral Evaluates to a Number 347 An Indefinite Integral Evaluates to a Function 348 Integration Is Inverse Differentiation 348 Every Infinite Series Has Two Related Sequences 349 Every Infinite Series Either Converges or Diverges 350 Chapter 17: Ten Tips to Take to the Test 351 Breathe 351 Start by Reading through the Exam 352 Solve the Easiest Problem First 352 Don’t Forget to Write dx and + C 352 Take the Easy Way Out Whenever Possible 352 If You Get Stuck, Scribble 353 If You Really Get Stuck, Move On 353 Check Your Answers 353 If an Answer Doesn’t Make Sense, Acknowledge It 354 Repeat the Mantra “I’m Doing My Best,” and Then Do Your Best 354 Index 355

About the Author :
Mark Zegarelli, a math tutor and writer with 25 years of professional experience, delights in making technical information crystal clear — and fun — for average readers. He is the author of Logic For Dummies and Basic Math & Pre-Algebra For Dummies.