Calculus II For Dummies E-Book

Mark Zegarelli

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch
• Veröffentlichungsjahr: 2023

Seitenzahl: 421

Beschreibung

The easy (okay, easier) way to master advanced calculus topics and theories Calculus II For Dummies will help you get through your (notoriously difficult) calc class--or pass a standardized test like the MCAT with flying colors. Calculus is required for many majors, but not everyone's a natural at it. This friendly book breaks down tricky concepts in plain English, in a way that you can understand. Practical examples and detailed walkthroughs help you manage differentiation, integration, and everything in between. You'll refresh your knowledge of algebra, pre-calc and Calculus I topics, then move on to the more advanced stuff, with plenty of problem-solving tips along the way. * Review Algebra, Pre-Calculus, and Calculus I concepts * Make sense of complicated processes and equations * Get clear explanations of how to use trigonometry functions * Walk through practice examples to master Calc II Use this essential resource as a supplement to your textbook or as refresher before taking a test--it's packed with all the helpful knowledge you need to succeed in Calculus II.

Leseprobe

Calculus II For Dummies®, 3rd Edition

Published by: John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, www.wiley.com

Copyright © 2023 by John Wiley & Sons, Inc., Hoboken, New Jersey

Published simultaneously in Canada

No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or 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 the prior written permission of the Publisher. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, [phone], fax [phone], or online at http://www.wiley.com/go/permissions.

Trademarks: Wiley, For Dummies, the Dummies Man logo, Dummies.com, Making Everything Easier, and related trade dress are trademarks or registered trademarks of John Wiley & Sons, Inc. and may not be used without written permission. All other trademarks are the property of their respective owners. John Wiley & Sons, Inc. is not associated with any product or vendor mentioned in this book.

LIMIT OF LIABILITY/DISCLAIMER OF WARRANTY: WHILE THE PUBLISHER AND AUTHORS HAVE USED THEIR BEST EFFORTS IN PREPARING THIS WORK, THEY MAKE NO REPRESENTATIONS OR WARRANTIES WITH RESPECT TO THE ACCURACY OR COMPLETENESS OF THE CONTENTS OF THIS WORK AND SPECIFICALLY DISCLAIM ALL WARRANTIES, INCLUDING WITHOUT LIMITATION ANY IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. NO WARRANTY MAY BE CREATED OR EXTENDED BY SALES REPRESENTATIVES, WRITTEN SALES MATERIALS OR PROMOTIONAL STATEMENTS FOR THIS WORK. THE FACT THAT AN ORGANIZATION, WEBSITE, OR PRODUCT IS REFERRED TO IN THIS WORK AS A CITATION AND/OR POTENTIAL SOURCE OF FURTHER INFORMATION DOES NOT MEAN THAT THE PUBLISHER AND AUTHORS ENDORSE THE INFORMATION OR SERVICES THE ORGANIZATION, WEBSITE, OR PRODUCT MAY PROVIDE OR RECOMMENDATIONS IT MAY MAKE. THIS WORK IS SOLD WITH THE UNDERSTANDING THAT THE PUBLISHER IS NOT ENGAGED IN RENDERING PROFESSIONAL SERVICES. THE ADVICE AND STRATEGIES CONTAINED HEREIN MAY NOT BE SUITABLE FOR YOUR SITUATION. YOU SHOULD CONSULT WITH A SPECIALIST WHERE APPROPRIATE. FURTHER, READERS SHOULD BE AWARE THAT WEBSITES LISTED IN THIS WORK MAY HAVE CHANGED OR DISAPPEARED BETWEEN WHEN THIS WORK WAS WRITTEN AND WHEN IT IS READ. NEITHER THE PUBLISHER NOR AUTHORS SHALL BE LIABLE FOR ANY LOSS OF PROFIT OR ANY OTHER COMMERCIAL DAMAGES, INCLUDING BUT NOT LIMITED TO SPECIAL, INCIDENTAL, CONSEQUENTIAL, OR OTHER DAMAGES.

For general information on our other products and services, please contact our Customer Care Department within the U.S. at 877-762-2974, outside the U.S. at 317-572-3993, or fax [phone]. For technical support, please visit https://hub.wiley.com/community/support/dummies.

Wiley publishes in a variety of print and electronic formats and by print-on-demand. Some material included with standard print versions of this book may not be included in e-books or in print-on-demand. If this book refers to media such as a CD or DVD that is not included in the version you purchased, you may download this material at http://booksupport.wiley.com. For more information about Wiley products, visit www.wiley.com.

Library of Congress Control Number: 2023930772

ISBN 978-1-119-98661-4 (pbk); ISBN 978-1-119-98662-1 (ebk); ISBN 978-1-119-98663-8 (ebk)

Calculus II For Dummies®

To view this book's Cheat Sheet, simply go to www.dummies.com and search for “Calculus II For Dummies Cheat Sheet” in the Search box.

Table of Contents

Cover

Title Page

Copyright

Introduction

About This Book

Conventions Used in This Book

What You're Not to Read

Foolish Assumptions

Icons Used in This Book

Beyond the Book

Where to Go from Here

Part 1: Introduction to Integration

Chapter 1: An Aerial View of the Area Problem

Checking Out the Area

Slicing Things Up

Defining the Indefinite

Solving Problems with Integration

Differential Equations

Understanding Infinite Series

Chapter 2: Forgotten but Not Gone: Review of Algebra and Pre-Calculus

Quick Review of Pre-Algebra and Algebra

Review of Pre-Calculus

Chapter 3: Recent Memories: Review of Calculus I

Knowing Your Limits

Hitting the Slopes with Derivatives

Understanding Differentiation

Finding Limits Using L’Hôpital’s Rule

Part 2: From Definite to Indefinite Integrals

Chapter 4: Approximating Area with Riemann Sums

Three Ways to Approximate Area with Rectangles

Two More Ways to Approximate Area

Building the Riemann Sum Formula

Chapter 5: There Must Be a Better Way — Introducing the Indefinite Integral

FTC2: The Saga Begins

Your New Best Friend: The Indefinite Integral

FTC1: The Journey Continues

Part 3: Evaluating Indefinite Integrals

Chapter 6: Instant Integration: Just Add Water (And

C

)

Evaluating Basic Integrals

Evaluating More Difficult Integrals

Understanding Integrability

Chapter 7: Sharpening Your Integration Moves

Integrating Rational and Radical Functions

Using Algebra to Integrate Using the Power Rule

Integrating Trig Functions

Integrating Compositions of Functions with Linear Inputs

Chapter 8: Here’s Looking at

U

-Substitution

Knowing How to Use U-Substitution

Recognizing When to Use U-Substitution

Using Substitution to Evaluate Definite Integrals

Part 4: Advanced Integration Techniques

Chapter 9: Parting Ways: Integration by Parts

Introducing Integration by Parts

Integrating by Parts with the DI-agonal Method

Chapter 10: Trig Substitution: Knowing All the (Tri)Angles

Integrating the Six Trig Functions

Integrating Powers of Sines and Cosines

Integrating Powers of Tangents and Secants

Integrating Powers of Cotangents and Cosecants

Integrating Weird Combinations of Trig Functions

Using Trig Substitution

Chapter 11: Rational Solutions: Integration with Partial Fractions

Strange but True: Understanding Partial Fractions

Solving Integrals by Using Partial Fractions

Beyond the Four Cases: Knowing How to Set Up Any Partial Fraction

Integrating Improper Rationals

Part 5: Applications of Integrals

Chapter 12: Forging into New Areas: Solving Area Problems

Breaking Us in Two

Improper Integrals

Finding the Unsigned Area of Shaded Regions on the

xy

-Graph

The Mean Value Theorem for Integrals

Calculating Arc Length

Chapter 13: Pump Up the Volume: Using Calculus to Solve 3-D Problems

Slicing Your Way to Success

Turning a Problem on Its Side

Two Revolutionary Problems

Finding the Space Between

Playing the Shell Game

Knowing When and How to Solve 3-D Problems

Chapter 14: What’s So Different about Differential Equations?

Basics of Differential Equations

Solving Differential Equations

Part 6: Infinite Series

Chapter 15: Following a Sequence, Winning the Series

Introducing Infinite Sequences

Introducing Infinite Series

Getting Comfy with Sigma Notation

Connecting a Series with Its Two Related Sequences

Recognizing Geometric Series and p-Series

Chapter 16: Where Is This Going? Testing for Convergence and Divergence

Starting at the Beginning

Using the

n

th-Term Test for Divergence

Let Me Count the Ways

Choosing Comparison Tests

Two-Way Tests for Convergence and Divergence

Looking at Alternating Series

Chapter 17: Dressing Up Functions with the Taylor Series

Elementary Functions

Power Series: Polynomials on Steroids

Expressing Functions as Series

Introducing the Maclaurin Series

Introducing the Taylor Series

Understanding Why the Taylor Series Works

Part 7: The Part of Tens

Chapter 18: Ten “Aha!” Insights in Calculus II

Integrating Means Finding the Area

When You Integrate, Area Means Signed Area

Integrating Is Just Fancy Addition

Integration Uses Infinitely Many Infinitely Thin Slices

Integration Contains a Slack Factor

A Definite Integral Evaluates to a Number

An Indefinite Integral Evaluates to a Function

Integration Is Inverse Differentiation

Every Infinite Series Has Two Related Sequences

Every Infinite Series Either Converges or Diverges

Chapter 19: Ten Tips to Take to the Test

Breathe

Start by Doing a Memory Dump as You Read through the Exam

Solve the Easiest Problem First

Don’t Forget to Write

dx

and

+ C

Take the Easy Way Out Whenever Possible

If You Get Stuck, Scribble

If You Really Get Stuck, Move On

Check Your Answers

If an Answer Doesn’t Make Sense, Acknowledge It

Repeat the Mantra, “I’m Doing My Best,” and Then Do Your Best

Index

About the Author

Advertisement Page

Connect with Dummies

End User License Agreement

List of Tables

Chapter 2

TABLE 2-1 Positive and Negative Integer Exponents of 2

TABLE 2-2 Rules for Simplifying Exponents

TABLE 2-3 Five Vertical and Five Horizontal Transformations of Functions

Chapter 3

TABLE 3-1 Approximating

TABLE 3-2 Cases of Indeterminate Forms Where You Can’t Apply L’Hôpital’s rule Di...

Chapter 4

TABLE 4-1 Approximating Area by Using Trapezoids

Chapter 6

TABLE 6-1 The 17 Basic Integrals (Antiderivatives)

Chapter 7

TABLE 7-1 Functions That Anti-differentiate to Inverse Trig Functions

TABLE 7-2 Anti-differentiating the six basic trig functions

TABLE 7-3 Formulas for anti-differentiating the six basic trig functions with li...

Chapter 9

TABLE 9-1 Knowing When to Integrate by Parts

Chapter 10

TABLE 10-1 Expressing the Six Trig Functions as a Pair of Trig Functions

TABLE 10-2 The Three Trig Substitution Cases

Chapter 11

TABLE 11-1 The Four Cases for Setting Up Partial Fractions

Chapter 15

TABLE 15-1 Infinite Sequences versus Infinite Series

TABLE 15-2 Series and Their Two Related Sequences

Chapter 16

TABLE 16-1 Understanding Absolute and Conditional Convergence of Alternating Ser...

Chapter 17

TABLE 17-1 Approximating the Value of sin 3

TABLE 17-2 Approximating the Value of sin 10

Guide

Cover

Title Page

Copyright

Table of Contents

Begin Reading

Index

About the Author

Pages

iii

iv

1

2

3

4

5

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

101

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

141

142

143

144

145

146

147

148

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

176

177

178

179

180

181

182

183

184

185

186

187

188

189

190

191

192

193

194

195

197

198

199

200

201

202

203

204

205

206

207

208

209

210

211

212

213

214

215

216

217

219

220

221

222

223

224

225

226

227

228

229

230

231

232

233

234

235

236

237

238

239

240

241

242

243

244

245

246

247

248

249

250

251

252

253

254

255

256

257

258

259

260

261

262

263

264

265

266

267

268

269

270

271

272

273

275

276

277

278

279

280

281

282

283

284

285

286

287

288

289

290

291

292

293

294

295

296

297

298

299

300

301

302

303

304

305

306

307

308

309

310

311

312

313

314

315

316

317

319

320

321

322

323

324

325

326

327

328

329

330

331

332

333

334

335

336

337

338

339

340

341

342

343

344

345

346

347

348

349

351

352

353

354

355

357

358

359

360

361

362

363

364

365

366

367

368

369

370

371

372

373

374

375

377

378

379

380

Introduction

Calculus is the great Mount Everest of math. Most of the world is content to just gaze upward at it in awe. But only a few brave souls attempt the ascent.

Or maybe not.

In recent years, calculus has become a required course not only for math, engineering, and physics majors, but also for students of biology, economics, psychology, nursing, and business. Law schools and MBA programs welcome students who’ve taken calculus because it demonstrates discipline and clarity of mind. High schools now have multiple math tracks that include calculus, from the basic college prep track to the AP tracks that prepare students for the Advanced Placement exam.

So perhaps calculus is more like a well-traveled Vermont mountain, with lots of trails and camping spots, plus a big ski lodge on top. You may need some stamina to conquer it, but with the right guide (this book, for example!), you’re not likely to find yourself swallowed up by a snowstorm half a mile from the summit.

About This Book

You can learn calculus. That’s what this book is all about. In fact, as you read these words, you may well already be a winner, having passed a course in Calculus I. If so, then congratulations and a nice pat on the back are in order.

Having said that, I want to discuss a few rumors you may have heard about Calculus II:

Calculus II is harder than Calculus I.

Calculus II is harder, even, than either Calculus III or Differential Equations.

Calculus II is more frightening than having your home invaded by zombies in the middle of the night and will result in emotional trauma requiring years of costly psychotherapy to heal.

Now, I admit that Calculus II is harder than Calculus I. Also, I may as well tell you that many — but not all — math students find it to be harder than the two semesters of math that follow. (Speaking personally, I found Calc II to be easier than Differential Equations.) But I’m holding my ground that the long-term psychological effects of a zombie attack far outweigh those awaiting you in any one-semester math course.

The two main topics of Calculus II are integration and infinite series. Integration is the inverse of differentiation, which you study in Calculus I. (For practical purposes, integration is a method for finding the area of unusual geometric shapes.) An infinite series is a sum of numbers that goes on forever, like 1 + 2 + 3 + … or … . Roughly speaking, most teachers focus on integration for the first two-thirds of the semester and infinite series for the last third.

This book gives you a solid introduction to what’s covered in a college course in Calculus II. You can use it either for self-study or while enrolled in a Calculus II course.

So feel free to jump around. Whenever I cover a topic that requires information from earlier in the book, I refer you to that section in case you want to refresh yourself on the basics.

Here are two pieces of advice for math students (remember them as you read the book):

Study a little every day. I know that students face a great temptation to let a book sit on the shelf until the night before an assignment is due. This is a particularly poor approach for Calc II. Math, like water, tends to seep in slowly and swamp the unwary!

So, when you receive a homework assignment, read over every problem as soon as you can and try to solve the easy ones. Go back to the harder problems every day, even if it’s just to reread and think about them. You’ll probably find that over time, even the most opaque problem starts to make sense.

Use practice problems for practice.

After you read through an example and think you understand it, copy the problem down on paper, close the book, and try to work it through. If you can get through it from beginning to end, you’re ready to move on. If not, go ahead and peek, but then try solving the problem later without peeking. (Remember, on exams, no peeking is allowed!)

Conventions Used in This Book

Throughout the book, I use the following conventions:

Italicized

text highlights new words and defined terms.

Boldfaced

text indicates keywords in bulleted lists and the action parts of numbered steps.

Monofont

text highlights web addresses.

Angles are measured in radians rather than degrees, unless I specifically state otherwise. (See

Chapter 2

for a discussion about the advantages of using radians for measuring angles.)

What You're Not to Read

All authors believe that each word they write is pure gold, but you don’t have to read every word in this book unless you really want to. You can skip over sidebars (those gray shaded boxes) where I go off on a tangent, unless you find that tangent interesting. Also feel free to pass by paragraphs labeled with the Technical Stuff icon.

If you’re not taking a class where you’ll be tested and graded, you can skip paragraphs labeled with the Tip icon and jump over extended step-by-step examples. However, if you’re taking a class, read this material carefully and practice working through examples on your own.

Foolish Assumptions

Not surprisingly, a lot of Calculus II builds on topics introduced in Calculus I and Pre-Calculus. So here are the foolish assumptions I make about you as you begin to read this book:

If you’re a student in a Calculus II course, I assume that you passed Calculus I. (Even if you got a D-minus, your Calc I professor and I agree that you’re good to go!)

If you’re studying on your own, I assume that you’re at least passably familiar with some of the basics of Calculus I.

I expect that you know some things from Calculus I, Algebra, and even Pre-Algebra, but I don’t throw you in the deep end of the pool and expect you to swim or drown. Chapter 2 contains a ton of useful Algebra and Pre-Algebra tidbits that you may have missed the first time around. And in Chapter 3, I give you a review of the most important topics from Calculus I that you’re sure to need in Calculus II. Furthermore, throughout the book, whenever I introduce a topic that calls for previous knowledge, I point you to an earlier chapter or section so you can get a refresher.

Icons Used in This Book

Here are four useful icons to help you navigate your way through the book:

Tips are helpful hints that show you the easy way to get things done. Try them out, especially if you’re taking a math course.

This icon points out key ideas that you need to know. Make sure you understand these ideas before reading on.

This icon points out interesting trivia that you can read or skip over as you like.

Warnings flag common errors that you want to avoid. Get clear where these traps are hiding so you don’t fall in.

Examples walk you through a particular math exercise designed to illustrate a particular topic. Practice makes perfect!

Beyond the Book

In addition to the introduction you’re reading right now, this book comes with a free, access-anywhere Cheat Sheet containing information worth remembering about Calculus II. To get this Cheat Sheet, simply go to www.dummies.com and type Calculus II For Dummies Cheat Sheet in the Search box.

Where to Go from Here

You can use this book either for self-study or to help you survive and thrive in a course in Calculus II.

If you’re taking a Calculus II course, you may be under pressure to complete a homework assignment or study for an exam. In that case, feel free to skip right to the topic that you need help with. Every section is self-contained, so you can jump right in and use the book as a handy reference. And when I refer to information that I discuss earlier in the book, I give you a brief review and a pointer to the chapter or section where you can get more information if you need it.

If you’re studying on your own, I recommend that you begin with Chapter 1, where I give you an overview of the entire book, and then read the chapters from beginning to end. Jump over Chapters 2 and 3 if you feel confident about your grounding in the math leading up to Calculus II. And, of course, if you’re dying to read about a topic that’s later in the book, go for it! You can always drop back to an easier chapter if you get lost.

Part 1

Introduction to Integration

IN THIS PART …

See Calculus II as an ordered approach to finding the area of unusual shapes on the

xy

-graph

Use the definite integral to clearly define an area problem

Slice an irregularly shaped area into rectangles to approximate area

Review the math you need from Pre-Algebra, Algebra, Pre-Calculus, and Calculus I