Topics

Advanced techniques of integration

    • Using integration by parts, possibly multiple times
    • Trigonometric integrals.
    • Using trigonometric substitution for integrals involving square roots and other related expressions.
    • Using partial fractions to work with rational functions.
    • Numerical integration using left hand, midpoint, right hand, trapezoid, and Simpson rules. Using error estimates of definite integral.
    • Improper integrals (involving ∞ as either a horizontal and/or vertical asymptote). Determining if an improper integral converges and giving appropriate bounds.
    • Combining techniques.

Applications of integration

    • Volumes by cross sections, volumes of revolution, using cylindrical shells.
    • Arc length of a curve (possibly parametric).
    • Surface area formed by revolution of a curve (possibly parametric).
    • Computing work done; Hooke's law; fluid forces on a plate.
    • Finding mass, moments, and center of mass of a weighted line segment and constant density regions in the plane.
    • Using parameterized curves.
    • Polar coordinate system.
    • Graphing in polar coordinates; area and lengths in polar coordinates.

Sequences and series

    • Understand sequences, and limits of sequences, e.g., L'Hospital's rule. Determine if a sequence is monotone.
    • Find the sum for geometric and telescoping series.
    • Know how to use the basic divergence test (or n-th term test), comparison test, limit comparison test, integral test, p-series test, ratio test, and root test to determine convergence of a series.
    • Absolute and conditional convergence of series. Alternating series test.
    • Power series, center of power series, radius of convergence, interval of convergence.
    • Integrating and differentiating power series.
    • Taylor polynomials and Taylor series, centered at a given point.
    • Use power series to compute higher order derivatives. Use Taylor series to compute integrals.
    • Give Maclaurin series for sin(x), cos(x), e^x, 1/(1-x), and arctan(x).
    • Using error estimates for Taylor polynomials.
    • Binomial series.