Thomas’ Calculus Early Transcendentals 12ed

Study Guide for Chap 08

Coverage: Sections 8.1-8.4, 8.7.

1. List all techniques of integrations (don’t forget the ‘half-angle’ substitution z = tan(x/2)), create a few examples for each of them and then solve. Then go through problems 69-116 on page 509 as a final review.

2. Study and memorize the convergence/divergence of ∫_{0}^{a} x^{−p}dx and ∫_{a}^{∞} x^{−p}dx for the cases 0 < p < 1, p = 1 and p > 1, respectively. Here a is an arbitrary positive number. The exact value of a > 0 is not important (why?).

3. Which comparison Theorems can be used to decide convergence/divergence of improper integrals?

4. How to determine convergence/divergence of an improper integral if it has multiple ‘improper’ parts such as the domain contains both x → ±∞, and/or several points of discontinuity within the domain?