Calculus II For Dummies

Calculus II For Dummies
About the Author
Dedication
Author’s Acknowledgments
Contents at a Glance
Table of Contents
Introduction
About This Book
Conventions Used in This Book
What You’re Not to Read
Foolish Assumptions
How This Book Is Organized
Icons Used in This Book
Where to Go from Here
Part I: 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
Understanding Infinite Series
Advancing Forward into Advanced Math
Chapter 2: Dispelling Ghosts from the Past: A Review of Pre-Calculus and Calculus I
Forgotten but Not Gone: A Review of Pre-Calculus
Recent Memories: A Review of Calculus I
Finding Limits by Using L’Hospital’s Rule
Chapter 3: From Definite to Indefinite: The Indefinite Integral
Approximate Integration
Knowing Sum-Thing about Summation Formulas
As Bad as It Gets: Calculating Definite Integrals by Using the Riemann Sum Formula
Light at the End of the Tunnel: The Fundamental Theorem of Calculus
Understanding the Fundamental Theorem of Calculus
Your New Best Friend: The Indefinite Integral
Part II: Indefinite Integrals
Chapter 4: Instant Integration: Just Add Water (And C)
Evaluating Basic Integrals
Evaluating More Difficult Integrals
Understanding Integrability
Chapter 5: Making a Fast Switch: Variable Substitution
Knowing How to Use Variable Substitution
Recognizing When to Use Substitution
Using Substitution to Evaluate Definite Integrals
Chapter 6: Integration by Parts
Introducing Integration by Parts
Integrating by Parts with the DI-agonal Method
Chapter 7: 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 8: When All Else Fails: Integration with Partial Fractions
Strange but True: Understanding Partial Fractions
Solving Integrals by Using Partial Fractions
Integrating Improper Rationals
Part III: Intermediate Integration Topics
Chapter 9: Forging into New Areas: Solving Area Problems
Breaking Us in Two
Improper Integrals
Solving Area Problems with More Than One Function
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
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
Part IV: Infinite Series
Chapter 11: 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 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
Using Comparison Tests
Two-Way Tests for Convergence and Divergence
Alternating Series
Chapter 13: 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 V: Advanced Topics
Chapter 14: Multivariable Calculus
Visualizing Vectors
Leaping to Another Dimension
Functions of Several Variables
Partial Derivatives
Multiple Integrals
Chapter 15: What’s so Different about Differential Equations?
Basics of Differential Equations
Solving Differential Equations
Part VI: The Part of Tens
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
Index