Calculus II, Spring 2026 (http://www.math.nthu.edu.tw/˜ wangwc/) Thomas’ Calculus Early Transcendentals 13ed

Study guide for quiz 01

Quiz problems include both the lecture contents and homework problems.

1. Section 8.8:

Memorize the definitions of type I and type II improper integrals and study how to verify convergence/divergence of improper integrals from these definitions. You may need to review some techniques of integration, such as ”integration by parts”, etc..

2. Section 8.8:

Study the convergence/divergence of the improper integrals ∫_{a}^{∞} \frac{1}{x^{p}} dx (a > 0), ∫_{0}^{a} \frac{1}{x^{p}} dx,

∫_{c}^{c+a} \frac{1}{(x − c)^{p}} dx, and ∫_{c−a}^{c} \frac{1}{|x − c|^{p}} dx for p > 1, p = 1 and 0 < p < 1, respectively by

direct computation. Then memorize the results. Plotting the graph of f(x) = \frac{1}{x^{p}} for various p’s as in page 11 of lecture 01 would likely help you memorize the results.

3. Section 8.8: Study the comparison Theorems (Theorem 2, Theorem 3) used to decide convergence/divergence of improper integrals. Sample problems: section 8.8, problems 35-64.

4. Section 8.8: Study how to determine convergence/divergence of an improper integral with multiple ‘improper’ parts, such as the domain contains both x → ±∞, and/or points of discontinuity within the domain.