## Topics

• 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.