## Topics

Topics

### Advanced techniques of integration

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

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

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.