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Table of contents
  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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. 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
A
absolute convergence 5.5 Alternating Series
absolute error 3.6 Numerical Integration
Airy’s equation 6.4 Working with Taylor Series
algebraic function 3.1 Integration by Parts
alternating series 5.5 Alternating Series
alternating series test 5.5 Alternating Series
angular coordinate 7.3 Polar Coordinates
annuity payments 6 Review Exercises
Archimedean spiral 7.3 Polar Coordinates
area under the curve 1.1 Approximating Areas
arithmetic sequence 5.1 Sequences
asymptotically semi-stable solution 4.2 Direction Fields and Numerical Methods
asymptotically stable solution 4.2 Direction Fields and Numerical Methods
asymptotically unstable solution 4.2 Direction Fields and Numerical Methods
autonomous differential equation 4.3 Separable Equations
average value of the function 1.2 The Definite Integral
B
bounded above 5.1 Sequences
bounded below 5.1 Sequences
bounded sequence 5.1 Sequences
G
Gabriel’s Horn 3.7 Improper Integrals
general form 7.5 Conic Sections
geometric sequence 5.1 Sequences
geometric series 5.2 Infinite Series
golden ratio 5.1 Sequences
Gompertz equation 4.4 The Logistic Equation
I
improper integral 3.7 Improper Integrals
indefinite integrals 3.1 Integration by Parts
index variable 5.1 Sequences
infinite sequence 5.1 Sequences
infinite series 5.2 Infinite Series
initial population 4.4 The Logistic Equation
initial-value problem 4.1 Basics of Differential Equations
integrable function 1.2 The Definite Integral
integration by parts 3.1 Integration by Parts
integration by substitution 1.5 Substitution
interval of convergence 6.1 Power Series and Functions
K
Koch’s snowflake 5.2 Infinite Series
L
left-endpoint approximation 1.1 Approximating Areas
limit comparison test 5.4 Comparison Tests
limit of the sequence 5.1 Sequences
limits of integration 1.2 The Definite Integral
logarithmic function 3.1 Integration by Parts
logistic differential equation 4.4 The Logistic Equation
M
Maclaurin polynomials 6.3 Taylor and Maclaurin Series
major axis 7.5 Conic Sections
Mean Value Theorem for Integrals 1.3 The Fundamental Theorem of Calculus
method of cylindrical shells. 2.3 Volumes of Revolution: Cylindrical Shells
method of equating coefficients 3.4 Partial Fractions
method of exhaustion 1.1 Approximating Areas
method of strategic substitution 3.4 Partial Fractions
minor axis 7.5 Conic Sections
monotone sequence 5.1 Sequences
P
parameterization of a curve 7.1 Parametric Equations
parametric curve 7.1 Parametric Equations
parametric equations 7.1 Parametric Equations
partial fraction decomposition 3.4 Partial Fractions
partial sum 5.2 Infinite Series
Pascal’s principle 2.5 Physical Applications
polar coordinate system 7.3 Polar Coordinates
polar equations 7.3 Polar Coordinates
power reduction formulas 3.2 Trigonometric Integrals
power-reducing identities 3.2 Trigonometric Integrals
probability density function 3.7 Improper Integrals
prolate cycloid 7.1 Parametric Equations
T
Taylor polynomials 6.3 Taylor and Maclaurin Series
Taylor’s theorem with remainder 6.3 Taylor and Maclaurin Series
telescoping series 5.2 Infinite Series
term-by-term differentiation of a power series 6.2 Properties of Power Series
term-by-term integration of a power series 6.2 Properties of Power Series
theorem of Pappus for volume 2.6 Moments and Centers of Mass
threshold population 4.4 The Logistic Equation
traffic accidents 3.7 Improper Integrals
trapezoidal rule 3.6 Numerical Integration
trigonometric integrals 3.2 Trigonometric Integrals
trigonometric substitution 3.3 Trigonometric Substitution
U
unbounded sequence 5.1 Sequences
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