Calculus II For Dummies

by Mark Zegarelli

Overview

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 ForDummies offers expert instruction, advice, and tips to helpsecond semester calculus students get a handle on the subject andace 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, approximateintegration, and improper integrals. This hands-on guide alsocovers sequences and series, with introductions to multivariablecalculus, differential equations, and numerical analysis. Best ofall, it includes practical exercises designed to simplify andenhance 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? Thisplain-English guide to Calculus II will set you straight!

Product Details

ISBN-13:	9781118161708
Publisher:	Wiley
Publication date:	01/24/2012
Series:	For Dummies Books
Edition description:	2nd ed.
Pages:	384
Sales rank:	169,107
Product dimensions:	7.30(w) x 9.10(h) x 1.00(d)

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.

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