Contents

• Preface
  • Computers
  
  
• Definite and Indefinite Integrals
  • The Definite Integral
    • The definition of area under curve
    • Relation between velocity and area
    • Definition of Integral
    • The Fundamental Theorem of Calculus
  • Indefinite Integrals and Change
    • Indefinite Integrals
    • Examples
    • Physical Intuition
  • Substitution and Symmetry
    • The Substitution Rule
    • The Substitution Rule for Definite Integrals
    • Symmetry
  
  
• Applications to Areas, Volume, and Averages
  • Using Integration to Determine Areas Between Curves
    • Examples
  • Computing Volumes of Surfaces of Revolution
  • Average Values
  
  
• Polar Coordinates and Complex Numbers
  • Polar Coordinates
  • Areas in Polar Coordinates
    • Examples
  • Complex Numbers
    • Polar Form
  • Complex Exponentials and Trig Identities
    • Trigonometry and Complex Exponentials
  
  
• Integration Techniques
  • Integration By Parts
  • Trigonometric Integrals
    • Some Remarks on Using Complex-Valued Functions
  • Trigonometric Substitutions
  • Factoring Polynomials
  • Integration of Rational Functions Using Partial Fractions
  • Approximating Integrals
  • Improper Integrals
    • Convergence, Divergence, and Comparison
  
  
• Sequences and Series
  • Sequences
  • Series
  • The Integral and Comparison Tests
    • Estimating the Sum of a Series
  • Tests for Convergence
    • The Comparison Test
    • Absolute and Conditional Convergence
    • The Ratio Test
    • The Root Test
  • Power Series
    • Shift the Origin
    • Convergence of Power Series
  • Taylor Series
  • Applications of Taylor Series
    • Estimation of Taylor Series
  
  
• Some Differential Equations
  • Separable Equations
  • Logistic Equation
  
  
• Index