Table of Contents

About This Book

Conventions Used in This Book

What You’re Not to Read

Foolish Assumptions

How This Book Is Organized

Part I: Introduction to Integration

Part II: Indefinite Integrals

Part III: Intermediate Integration Topics

Part IV: Infinite Series

Part V: Advanced Topics

Part VI: The Part of Tens

Icons Used in This Book

Where to Go from Here

Checking Out the Area

Comparing classical and analytic geometry

Discovering a new area of study

Generalizing the area problem

Finding definite answers with the definite integral

Slicing Things Up

Untangling a hairy problem using rectangles

Building a formula for finding area

Defining the Indefinite

Solving Problems with Integration

We can work it out: Finding the area between curves

Walking the long and winding road

You say you want a revolution

Understanding Infinite Series

Distinguishing sequences and series

Evaluating series

Identifying convergent and divergent series

Advancing Forward into Advanced Math

Multivariable calculus

Differential equations

Fourier analysis

Numerical analysis

Forgotten but Not Gone: A Review of Pre-Calculus

Knowing the facts on factorials

Polishing off polynomials

Powering through powers (exponents)

Noting trig notation

Figuring the angles with radians

Graphing common functions

Asymptotes

Transforming continuous functions

Identifying some important trig identities

Polar coordinates

Summing up sigma notation

Recent Memories: A Review of Calculus I

Knowing your limits

Hitting the slopes with derivatives

Referring to the limit formula for derivatives

Knowing two notations for derivatives

Understanding differentiation

Finding Limits Using L’Hopital’s Rule

Understanding determinate and indeterminate forms of limits

Introducing L’Hopital’s Rule

Alternative indeterminate forms

Approximate Integration

Three ways to approximate area with rectangles

The slack factor

Two more ways to approximate area

Knowing Sum-Thing about Summation Formulas

The summation formula for counting numbers

The summation formula for square numbers

The summation formula for cubic numbers

As Bad as It Gets: Calculating Definite Integrals Using the Riemann Sum Formula

Plugging in the limits of integration

Expressing the function as a sum in terms of i and n

Calculating the sum

Solving the problem with a summation formula

Evaluating the limit

Light at the End of the Tunnel: The Fundamental Theorem of Calculus

Understanding the Fundamental Theorem of Calculus

What’s slope got to do with it?

Introducing the area function

Connecting slope and area mathematically

Seeing a dark side of the FTC

Your New Best Friend: The Indefinite Integral

Introducing anti-differentiation

Solving area problems without the Riemann sum formula

Understanding signed area

Distinguishing definite and indefinite integrals

Chapter 4: Instant Integration: Just Add Water (And C)

Evaluating Basic Integrals

Using the 17 basic anti-derivatives for integrating

Three important integration rules

What happened to the other rules?

Evaluating More Difficult Integrals

Integrating polynomials

Integrating rational expressions

Using identities to integrate trig functions

Understanding Integrability

Taking a look at two red herrings of integrability

Getting an idea of what integrable really means

Chapter 5: Making a Fast Switch: Variable Substitution

Knowing How to Use Variable Substitution

Finding the integral of nested functions

Determining the integral of a product

Integrating a function multiplied by a set of nested functions

Recognizing When to Use Substitution

Integrating nested functions

Knowing a shortcut for nested functions

Substitution when one part of a function differentiates to the other part

Using Substitution to Evaluate Definite Integrals

Chapter 6: Integration by Parts

Introducing Integration by Parts

Reversing the Product Rule

Knowing how to integrate by parts

Knowing when to integrate by parts

Integrating by Parts with the DI-agonal Method

Looking at the DI-agonal chart

Using the DI-agonal method

Chapter 7: Trig Substitution: Knowing All the (Tri)Angles

Integrating the Six Trig Functions

Integrating Powers of Sines and Cosines

Odd powers of sines and cosines

Even powers of sines and cosines

Integrating Powers of Tangents and Secants

Even powers of secants with tangents

Odd powers of tangents with secants

Odd powers of tangents without secants

Even powers of tangents without secants

Even powers of secants without tangents

Odd powers of secants without tangents

Even powers of tangents with odd powers of secants

Integrating Powers of Cotangents and Cosecants

Integrating Weird Combinations of Trig Functions

Using Trig Substitution

Distinguishing three cases for trig substitution

Integrating the three cases

Knowing when to avoid trig substitution

Chapter 8: When All Else Fails: Integration with Partial Fractions

Strange but True: Understanding Partial Fractions

Looking at partial fractions

Using partial fractions with rational expressions

Solving Integrals by Using Partial Fractions

Setting up partial fractions case by case

Knowing the ABCs of finding unknowns

Integrating partial fractions

Integrating Improper Rationals

Distinguishing proper and improper rational expressions

Recalling polynomial division

Trying out an example

Chapter 9: Forging into New Areas: Solving Area Problems

Breaking Us in Two

Improper Integrals

Getting horizontal

Going vertical

Solving Area Problems with More Than One Function

Finding the area under more than one function

Finding the area between two functions

Looking for a sign

Measuring unsigned area between curves with a quick trick

The Mean Value Theorem for Integrals

Calculating Arc Length

Chapter 10: Pump Up the Volume: Using Calculus to Solve 3-D Problems

Slicing Your Way to Success

Finding the volume of a solid with congruent cross sections

Finding the volume of a solid with similar cross sections

Measuring the volume of a pyramid

Measuring the volume of a weird solid

Turning a Problem on Its Side

Two Revolutionary Problems

Solidifying your understanding of solids of revolution

Skimming the surface of revolution

Finding the Space Between

Playing the Shell Game

Peeling and measuring a can of soup

Using the shell method

Knowing When and How to Solve 3-D Problems

Chapter 11: Following a Sequence, Winning the Series

Introducing Infinite Sequences

Understanding notations for sequences

Looking at converging and diverging sequences

Introducing Infinite Series

Getting Comfy with Sigma Notation

Writing sigma notation in expanded form

Seeing more than one way to use sigma notation

Discovering the Constant Multiple Rule for series

Examining the Sum Rule for series

Connecting a Series with Its Two Related Sequences

A series and its defining sequence

A series and its sequences of partial sums

Recognizing Geometric Series and P-Series

Getting geometric series

Pinpointing p-series

Chapter 12: Where Is This Going? Testing for Convergence and Divergence

Starting at the Beginning

Using the nth-Term Test for Divergence

Let Me Count the Ways

One-way tests

Two-way tests

Choosing Comparison Tests

Getting direct answers with the direct comparison test

Testing your limits with the limit comparison test

Two-Way Tests for Convergence and Divergence

Integrating a solution with the integral test

Rationally solving problems with the ratio test

Rooting out answers with the root test

Looking at Alternating Series

Eyeballing two forms of the basic alternating series

Making new series from old ones

Alternating series based on convergent positive series

Checking out the alternating series test

Understanding absolute and conditional convergence

Testing alternating series

Chapter 13: Dressing Up Functions with the Taylor Series

Elementary Functions

Knowing two drawbacks of elementary functions

Appreciating why polynomials are so friendly

Representing elementary functions as polynomials

Representing elementary functions as series

Power Series: Polynomials on Steroids

Integrating power series

Understanding the interval of convergence

Expressing Functions as Series

Expressing sin x as a series

Expressing cos x as a series

Introducing the Maclaurin Series

Introducing the Taylor Series

Computing with the Taylor series

Examining convergent and divergent Taylor series

Expressing functions versus approximating functions

Calculating error bounds for Taylor polynomials

Understanding Why the Taylor Series Works

Chapter 14: Multivariable Calculus

Visualizing Vectors

Understanding vector basics

Distinguishing vectors and scalars

Calculating with vectors

Leaping to Another Dimension

Understanding 3-D Cartesian coordinates

Using alternative 3-D coordinate systems

Functions of Several Variables

Partial Derivatives

Measuring slope in three dimensions

Evaluating partial derivatives

Multiple Integrals

Measuring volume under a surface

Evaluating multiple integrals

Chapter 15: What’s So Different about Differential Equations?

Basics of Differential Equations

Classifying DEs

Looking more closely at DEs

Solving Differential Equations

Solving separable equations

Solving initial-value problems (IVPs)

Using an integrating factor

Chapter 16: 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 17: Ten Tips to Take to the Test

Breathe

Start by Reading 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

Cheat Sheet