Calculus II

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