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  1. Preface
  2. 1 Integration
    1. Introduction
    2. 1.1 Approximating Areas
    3. 1.2 The Definite Integral
    4. 1.3 The Fundamental Theorem of Calculus
    5. 1.4 Integration Formulas and the Net Change Theorem
    6. 1.5 Substitution
    7. 1.6 Integrals Involving Exponential and Logarithmic Functions
    8. 1.7 Integrals Resulting in Inverse Trigonometric Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  3. 2 Applications of Integration
    1. Introduction
    2. 2.1 Areas between Curves
    3. 2.2 Determining Volumes by Slicing
    4. 2.3 Volumes of Revolution: Cylindrical Shells
    5. 2.4 Arc Length of a Curve and Surface Area
    6. 2.5 Physical Applications
    7. 2.6 Moments and Centers of Mass
    8. 2.7 Integrals, Exponential Functions, and Logarithms
    9. 2.8 Exponential Growth and Decay
    10. 2.9 Calculus of the Hyperbolic Functions
    11. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Chapter Review Exercises
  4. 3 Techniques of Integration
    1. Introduction
    2. 3.1 Integration by Parts
    3. 3.2 Trigonometric Integrals
    4. 3.3 Trigonometric Substitution
    5. 3.4 Partial Fractions
    6. 3.5 Other Strategies for Integration
    7. 3.6 Numerical Integration
    8. 3.7 Improper Integrals
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  5. 4 Introduction to Differential Equations
    1. Introduction
    2. 4.1 Basics of Differential Equations
    3. 4.2 Direction Fields and Numerical Methods
    4. 4.3 Separable Equations
    5. 4.4 The Logistic Equation
    6. 4.5 First-order Linear Equations
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  6. 5 Sequences and Series
    1. Introduction
    2. 5.1 Sequences
    3. 5.2 Infinite Series
    4. 5.3 The Divergence and Integral Tests
    5. 5.4 Comparison Tests
    6. 5.5 Alternating Series
    7. 5.6 Ratio and Root Tests
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Chapter Review Exercises
  7. 6 Power Series
    1. Introduction
    2. 6.1 Power Series and Functions
    3. 6.2 Properties of Power Series
    4. 6.3 Taylor and Maclaurin Series
    5. 6.4 Working with Taylor Series
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Chapter Review Exercises
  8. 7 Parametric Equations and Polar Coordinates
    1. Introduction
    2. 7.1 Parametric Equations
    3. 7.2 Calculus of Parametric Curves
    4. 7.3 Polar Coordinates
    5. 7.4 Area and Arc Length in Polar Coordinates
    6. 7.5 Conic Sections
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  9. A | Table of Integrals
  10. B | Table of Derivatives
  11. C | Review of Pre-Calculus
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
  13. Index
absolute error
if BB is an estimate of some quantity having an actual value of A,A, then the absolute error is given by |AB||AB|
computer algebra system (CAS)
technology used to perform many mathematical tasks, including integration
improper integral
an integral over an infinite interval or an integral of a function containing an infinite discontinuity on the interval; an improper integral is defined in terms of a limit. The improper integral converges if this limit is a finite real number; otherwise, the improper integral diverges
integration by parts
a technique of integration that allows the exchange of one integral for another using the formula udv=uvvduudv=uvvdu
integration table
a table that lists integration formulas
midpoint rule
a rule that uses a Riemann sum of the form Mn=i=1nf(mi)Δx,Mn=i=1nf(mi)Δx, where mimi is the midpoint of the ith subinterval to approximate abf(x)dxabf(x)dx
numerical integration
the variety of numerical methods used to estimate the value of a definite integral, including the midpoint rule, trapezoidal rule, and Simpson’s rule
partial fraction decomposition
a technique used to break down a rational function into the sum of simple rational functions
power reduction formula
a rule that allows an integral of a power of a trigonometric function to be exchanged for an integral involving a lower power
relative error
error as a percentage of the absolute value, given by |ABA|=|ABA|·100%|ABA|=|ABA|·100%
Simpson’s rule
a rule that approximates abf(x)dxabf(x)dx using the integrals of a piecewise quadratic function. The approximation SnSn to abf(x)dxabf(x)dx is given by Sn=Δx3(f(x0)+4f(x1)+2f(x2)+4f(x3)+2f(x4)+4f(x5)++2f(xn2)+4f(xn1)+f(xn))Sn=Δx3(f(x0)+4f(x1)+2f(x2)+4f(x3)+2f(x4)+4f(x5)++2f(xn2)+4f(xn1)+f(xn)) trapezoidal rule a rule that approximates abf(x)dxabf(x)dx using trapezoids
trigonometric integral
an integral involving powers and products of trigonometric functions
trigonometric substitution
an integration technique that converts an algebraic integral containing expressions of the form a2x2,a2x2, a2+x2,a2+x2, or x2a2x2a2 into a trigonometric integral
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