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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
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
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
Comparing classical and analytic geometry Discovering a new area of study Generalizing the area problem Finding definite answers with the definite integral
Untangling a hairy problem using rectangles Building a formula for finding area
We can work it out: Finding the area between curves Walking the long and winding road You say you want a revolution
Distinguishing sequences and series Evaluating series Identifying convergent and divergent series
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
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
Knowing your limits Hitting the slopes with derivatives Referring to the limit formula for derivatives Knowing two notations for derivatives Understanding differentiation
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
Three ways to approximate area with rectangles The slack factor Two more ways to approximate area
The summation formula for counting numbers The summation formula for square numbers The summation formula for cubic numbers
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
What’s slope got to do with it? Introducing the area function Connecting slope and area mathematically Seeing a dark side of the FTC
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
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
Using the 17 basic anti-derivatives for integrating Three important integration rules What happened to the other rules?
Integrating polynomials Integrating rational expressions Using identities to integrate trig functions
Taking a look at two red herrings of integrability Getting an idea of what integrable really means
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
Finding the integral of nested functions Determining the integral of a product Integrating a function multiplied by a set of nested functions
Integrating nested functions Knowing a shortcut for nested functions Substitution when one part of a function differentiates to the other part
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
Reversing the Product Rule Knowing how to integrate by parts Knowing when to integrate by parts
Looking at the DI-agonal chart Using the DI-agonal method
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
Odd powers of sines and cosines Even powers of sines and cosines
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
Distinguishing three cases for trig substitution Integrating the three cases Knowing when to avoid trig substitution
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
Looking at partial fractions Using partial fractions with rational expressions
Setting up partial fractions case by case Knowing the ABCs of finding unknowns Integrating partial fractions
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
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
Getting horizontal Going vertical
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
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
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
Solidifying your understanding of solids of revolution Skimming the surface of revolution
Peeling and measuring a can of soup Using the shell method
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
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
Understanding notations for sequences Looking at converging and diverging sequences
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
A series and its defining sequence A series and its sequences of partial sums
Getting geometric series Pinpointing p-series
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
One-way tests Two-way tests
Getting direct answers with the direct comparison test Testing your limits with the limit comparison test
Integrating a solution with the integral test Rationally solving problems with the ratio test Rooting out answers with the root test
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
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
Knowing two drawbacks of elementary functions Appreciating why polynomials are so friendly Representing elementary functions as polynomials Representing elementary functions as series
Integrating power series Understanding the interval of convergence
Expressing sin x as a series Expressing cos x as a series
Computing with the Taylor series Examining convergent and divergent Taylor series Expressing functions versus approximating functions Calculating error bounds for Taylor polynomials
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
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
Understanding vector basics Distinguishing vectors and scalars Calculating with vectors
Understanding 3-D Cartesian coordinates Using alternative 3-D coordinate systems
Measuring slope in three dimensions Evaluating partial derivatives
Measuring volume under a surface Evaluating multiple integrals
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
Classifying DEs Looking more closely at DEs
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
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
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
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