Improper Integrals

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 7.8 – Improper Integrals

Expected Educational Results

• Objective 09–01: I can identify an improper integral.

• Objective 09–02: I can use limits to evaluate an improper integral.

• Objective 09–03: I can use the Comparison Theorem to determine if an improper integral converges or diverges.

Bloom’s Taxonomy

A modern version of Bloom’s Taxonomy is included here to recognize various different levels of understanding and to encourage you to work towards higher-order understanding (those at the top of the pyramid). All Objectives, Investigations, Activities, etc. are color-coded with the level of understanding.

Real Numbers

Definition: Real Numbers

The set of Real Numbers is denoted by $\mathbb{R}$$\mathbb{R}$. The set of Real Numbers is the open set, $\mathbb{R}=(-\mathrm{\infty },\mathrm{\infty })$$\mathbb{R}=\left(-\infty, \infty\right)$.

This open set does NOT contain $-\mathrm{\infty }$$-\infty$ and $\mathrm{\infty }$$\infty$. In other words, $\pm \mathrm{\infty }$$\pm\infty$ are not Real Numbers.

NOTE: There exists the Extended Real Number System; but we will not use ${\mathbb{R}}^{\mathrm{\#}}=\mathbb{R}\cup \{-\mathrm{\infty },\mathrm{\infty }\}=[-\mathrm{\infty },\mathrm{\infty }]$$\mathbb{R}^{\#}=\mathbb{R}\cup\{-\infty, \infty\}=[-\infty, \infty]$ in this course.

NOTE: This course, Calculus II, uses ONLY Real Numbers, i.e., will use neither $-\mathrm{\infty }$$-\infty$ nor $\mathrm{\infty }$$\infty$ as Real-numbered values.

Use of Infinity

Investigation 01

Which of the following are not valid for the set of real numbers, $\mathbb{R}$$\mathbb{R}$? In other words, which of the following use $\mathrm{\infty }$$\infty$ as a real number? Explain.

1. ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}{e}^x=\mathrm{\infty }}$$\displaystyle \lim_{x\rightarrow\infty}{e^x}=\infty$

2. ${\displaystyle \underset{x\to -\mathrm{\infty }}{lim}{e}^x=e^{-\mathrm{\infty }}=0}$$\displaystyle \lim_{x\rightarrow-\infty}{e^x}=e^{-\infty}=0$

3. ${\displaystyle \underset{x\to -\mathrm{\infty }}{lim}\frac{1}{x}=0}$$\displaystyle \lim_{x\rightarrow-\infty}{\frac{1}{x}}=0$

4. ${\displaystyle \underset{x\to 0^{+}}{lim}\frac{1}{x}=\mathrm{\infty }}$$\displaystyle \lim_{x\rightarrow 0^+}{\frac{1}{x}}=\infty$

CC BY-NC-SA 4.0

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

Created: Thursday, 17 September 2020 2:48 EDT Last Modified: Sunday, 12 June 2022 - 17:28 (EDT)