Calculus II For Dummies

Title Page
Copyright Page
Table of Contents
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
Comparing classical and analytic geometry
Finding definite answers with the definite integral
Slicing Things Up
Untangling a hairy problem using rectangles
Moving left, right, or center
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
Differential Equations
Understanding Infinite Series
Distinguishing sequences and series
Evaluating series
Identifying convergent and divergent series
Chapter 2 Forgotten but Not Gone: Review of Algebra and Pre-Calculus
Quick Review of Pre-Algebra and Algebra
Working with fractions
Adding fractions
Subtracting fractions
Multiplying fractions
Dividing fractions
Knowing the facts on factorials
Polishing off polynomials
Powering through powers (exponents)
Understanding zero and negative exponents
Understanding fractional exponents
Expressing functions using exponents
Rewriting rational functions using exponents
Simplifying rational expressions by factoring
Review of Pre-Calculus
Trigonometry
Noting trig notation
Figuring the angles with radians
Identifying some important trig identities
Asymptotes
Graphing common parent functions
Linear and polynomial functions
Exponential and logarithmic functions
Trigonometric functions
Transforming continuous functions
Polar coordinates
Summing up sigma notation
Chapter 3 Recent Memories: Review of Calculus I
Knowing Your Limits
Telling functions and limits apart
Evaluating limits
Hitting the Slopes with Derivatives
Referring to the limit formula for derivatives
Knowing two notations for derivatives
Understanding Differentiation
Memorizing key derivatives
Derivatives of the trig functions
Derivatives of the inverse trig functions
The Power rule
The Sum rule
The Constant Multiple rule
The Product rule
The Quotient rule
Evaluating functions from the inside out
Differentiating functions from the outside in
Finding Limits Using L’Hôpital’s Rule
Introducing L’Hôpital’s rule
Alternative indeterminate forms
Case #1: 0 ⋅ ∞
Case #2: ∞ – ∞
Case #3: Indeterminate powers
Part 2 From Definite to Indefinite Integrals
Chapter 4 Approximating Area with Riemann Sums
Three Ways to Approximate Area with Rectangles
Using left rectangles
Using right rectangles
Finding a middle ground: The Midpoint rule
Two More Ways to Approximate Area
Feeling trapped? The Trapezoid rule
Don’t have a cow! Simpson’s rule
Building the Riemann Sum Formula
Approximating the definite integral with the area formula for a rectangle
Widening your understanding of width
Limiting the margin of error
Summing things up with sigma notation
Heightening the functionality of height
Finishing with the slack factor
Chapter 5 There Must Be a Better Way — Introducing the Indefinite Integral
FTC2: The Saga Begins
Introducing FTC2
Evaluating definite integrals using FTC2
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
FTC1: The Journey Continues
Understanding area functions
Making sense of FTC1
Part 3 Evaluating Indefinite Integrals
Chapter 6 Instant Integration: Just Add Water (And C )
Evaluating Basic Integrals
Using the 17 basic antiderivatives for integrating
Three important integration rules
The Sum rule for integration
The Constant Multiple rule for integration
The Power rule for integration
What happened to the other rules?
Evaluating More Difficult Integrals
Integrating polynomials
Integrating more complicated-looking functions
Understanding Integrability
Taking a look at two red herrings of integrability
Computing integrals
Representing integrals as elementary functions
Getting an idea of what integrable really means
Chapter 7 Sharpening Your Integration Moves
Integrating Rational and Radical Functions
Integrating simple rational functions
Integrating radical functions
Using Algebra to Integrate Using the Power Rule
Integrating by using inverse trig functions
Integrating Trig Functions
Recalling how to anti-differentiate the six basic trig functions
Using the Basic Five trig identities
Applying the Pythagorean trig identities
Using  to integrate trig functions
Using  to integrate trig functions
Using  to integrate trig functions
Integrating Compositions of Functions with Linear Inputs
Understanding how to integrate familiar functions that have linear inputs
Integrating the  function composed with a linear input
Integrating the six basic trig functions with linear inputs
Integrating power functions composed with a linear input
Knowing the handy arctan formula
Using algebra to solve more complex problems
Using trig identities to integrate more complex functions
Understanding why integrating compositions of functions with linear inputs actually works
Chapter 8 Here’s Looking at U-Substitution
Knowing How to Use U-Substitution
Recognizing When to Use U-Substitution
The simpler case: f (x) · f ’(x)
The more complex case: g(  f (x)) · f ’(x) when you know how to integrate g (x)
Using Substitution to Evaluate Definite Integrals
Part 4 Advanced Integration Techniques
Chapter 9 Parting Ways: 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
L is for logarithm
I is for inverse trig
A is for algebraic
T is for trig
Chapter 10 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
Odd powers of tangents
Other tangent and secant cases
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
The sine case
The tangent case
The secant case
Knowing when to avoid trig substitution
Chapter 11 Rational Solutions: 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
Case 1: Distinct linear factors
Setting up partial fractions
Solving for unknowns A, B, and C
Evaluating the integral
Case 2: Repeated linear factors
Setting up partial fractions
Solving for unknowns A and B
Evaluating the integral
Case 3: Distinct quadratic factors
Setting up partial fractions
Solving for unknowns A, B, and C
Evaluating the integral
Case 4: Repeated quadratic factors
Setting up partial fractions
Solving for unknowns A, B, C, and D
Evaluating the integral
Beyond the Four Cases: Knowing How to Set Up Any Partial Fraction
Integrating Improper Rationals
Distinguishing proper and improper rational expressions
Trying out an example
Part 5 Applications of Integrals
Chapter 12 Forging into New Areas: Solving Area Problems
Breaking Us in Two
Improper Integrals
Getting horizontal
Going vertical
Handling asymptotic limits of integration
Piecing together discontinuous integrands
Finding the Unsigned Area of Shaded Regions on the xy-Graph
Finding unsigned area when a region is separated horizontally
Crossing the line to find unsigned area
Calculating the area under more than one function
Measuring a single shaded region between two functions
Finding the area of two or more shaded regions between two functions
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
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 without inverses
Knowing When and How to Solve 3-D Problems
Chapter 14 What’s So Different about Differential Equations?
Basics of Differential Equations
Classifying DEs
Ordinary and partial differential equations
Order of DEs
Linear DEs
Looking more closely at DEs
How every integral is a DE
Why building DEs is easier than solving them
Checking DE solutions
Solving Differential Equations
Solving separable equations
Solving initial-value problems
Part 6 Infinite Series
Chapter 15 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
Harmonizing with the harmonic series
Testing p-series when p = 2, p = 3, and p = 4
Testing p-series when 
Chapter 16 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 17 Dressing Up Functions with the Taylor Series
Elementary Functions
Identifying two drawbacks of elementary functions
Appreciating why polynomials are so friendly
Representing elementary functions as series
Power Series: Polynomials on Steroids
Integrating power series
Understanding the interval of convergence
The interval of convergence is never empty
Three varieties for 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
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
EULA