Math Ladder · College

Calculus 2: Integrals and Series

Techniques of integration, areas and volumes, improper integrals, parametric and polar curves, sequences and power series.

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45 topics in 6 units. Builds on Calculus 1: Limits and Derivatives. Leads to Calculus 3: Multivariable Calculus, Linear Algebra, Differential Equations, Probability, Numerical Analysis and Real Analysis. It also opens the way to E and M and Modern Physics on the Physics Ladder. Not sure this is the right course? Find your rung.

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Unit 1 Applications of Integration

10 topics

Find areas, volumes, arc lengths, work, fluid force and centers of mass.

  1. Area Between Curves
  2. Volume by Slicing
  3. Disks and Washers
  4. Cylindrical Shells
  5. Arc Length
  6. Surface Area of Revolution
  7. Work
  8. Hydrostatic Force
  9. Moments and Centers of Mass
  10. Hyperbolic Functions

Unit 2 Techniques of Integration

8 topics

Integrate by parts, with trig identities and substitutions, with partial fractions, and numerically.

  1. Integration by Parts
  2. Trigonometric Integrals
  3. Trigonometric Substitution
  4. Integration by Partial Fractions
  5. Choosing a Technique
  6. Trapezoid and Simpson's Rules
  7. Improper Integrals
  8. Comparison Test for Improper Integrals

Unit 3 First Differential Equations

4 topics

Meet differential equations, slope fields, Euler's method, separable equations and logistic growth.

  1. What a Differential Equation Is
  2. Direction Fields and Euler's Method
  3. Separable Equations
  4. The Logistic Equation

Unit 4 Parametric and Polar Calculus

3 topics

Find slopes, areas and lengths for parametric and polar curves.

  1. Tangents and Arc Length for Parametric Curves
  2. Area in Polar Coordinates
  3. Polar Tangents and Arc Length

Unit 5 Sequences and Series

12 topics

Decide whether sequences and series converge, and estimate sums.

  1. Sequences and Their Limits
  2. Monotone and Bounded Sequences
  3. Series and Partial Sums
  4. Geometric and Telescoping Series
  5. The Divergence Test
  6. The Integral Test and p-Series
  7. Comparison and Limit Comparison Tests
  8. Alternating Series
  9. Absolute and Conditional Convergence
  10. Ratio and Root Tests
  11. Choosing a Convergence Test
  12. Estimating the Sum of a Series

Unit 6 Power Series

8 topics

Represent functions as power series and Taylor series, and use them.

  1. Power Series and Intervals of Convergence
  2. Representing Functions as Power Series
  3. Differentiating and Integrating Power Series
  4. Taylor and Maclaurin Series
  5. Taylor's Inequality
  6. Working with Known Series
  7. The Binomial Series
  8. Applications of Series
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