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Calculus Volume 2

Key Equations

Calculus Volume 2Key Equations
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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

Key Equations

Integration by parts formula ∫udv=uv−∫vdu∫udv=uv−∫vdu
Integration by parts for definite integrals ∫abudv=uv|ab−∫abvdu∫abudv=uv|ab−∫abvdu

To integrate products involving sin(ax),sin(ax), sin(bx),sin(bx), cos(ax),cos(ax), and cos(bx),cos(bx), use the substitutions.

Sine Products sin(ax)sin(bx)=12cos((a−b)x)−12cos((a+b)x)sin(ax)sin(bx)=12cos((a−b)x)−12cos((a+b)x)
Sine and Cosine Products sin(ax)cos(bx)=12sin((a−b)x)+12sin((a+b)x)sin(ax)cos(bx)=12sin((a−b)x)+12sin((a+b)x)
Cosine Products cos(ax)cos(bx)=12cos((a−b)x)+12cos((a+b)x)cos(ax)cos(bx)=12cos((a−b)x)+12cos((a+b)x)
Power Reduction Formula ∫secnx dx=secn-2x tan xn−1+n−2n−1∫secn−2xdx;n≠1∫secnx dx=secn-2x tan xn−1+n−2n−1∫secn−2xdx;n≠1
Power Reduction Formula ∫tannxdx=1n−1tann−1x−∫tann−2xdx∫tannxdx=1n−1tann−1x−∫tann−2xdx
Midpoint rule Mn=∑i=1nf(mi)ΔxMn=∑i=1nf(mi)Δx
Trapezoidal rule Tn=12Δx(f(x0)+2f(x1)+2f(x2)+⋯+2f(xn−1)+f(xn))Tn=12Δx(f(x0)+2f(x1)+2f(x2)+⋯+2f(xn−1)+f(xn))
Simpson’s rule Sn=Δx3(f(x0)+4f(x1)+2f(x2)+4f(x3)+2f(x4)+4f(x5)+⋯+2f(xn−2)+4f(xn−1)+f(xn))Sn=Δx3(f(x0)+4f(x1)+2f(x2)+4f(x3)+2f(x4)+4f(x5)+⋯+2f(xn−2)+4f(xn−1)+f(xn))
Error bound for midpoint rule Error inMn≤M(b−a)324n2Error inMn≤M(b−a)324n2
Error bound for trapezoidal rule Error inTn≤M(b−a)312n2Error inTn≤M(b−a)312n2
Error bound for Simpson’s rule Error inSn≤M(b−a)5180n4Error inSn≤M(b−a)5180n4
Improper integrals ∫a+∞f(x)dx=limt→+∞∫atf(x)dx∫−∞bf(x)dx=limt→−∞∫tbf(x)dx∫−∞+∞f(x)dx=∫−∞0f(x)dx+∫0+∞f(x)dx∫a+∞f(x)dx=limt→+∞∫atf(x)dx∫−∞bf(x)dx=limt→−∞∫tbf(x)dx∫−∞+∞f(x)dx=∫−∞0f(x)dx+∫0+∞f(x)dx
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