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Calculus III

Course: Calculus II
Topic: Calculus of Infinity
Subtopic: Improper Integrals

Overview

Proper integrals are definite integrals that have real numbers as limits of integration. However, when one (or both) of the limits is infinity then it is an improper integral. In this case we must replace the infinity with a variable and look at the limit of the integral as that variable approaches infinity. The second type of improper integrals occurs when the function being integrated has a domain restriction on the interval [lower limit of integration, upper limit of integration]. In this case we must rewrite the integral as the sum of two integrals and use limits to approach the domain restriction.

Objectives

By the end of this topic you should know and be prepared to be tested on:

• 9.2.1 Rewrite an improper integral that has infinity as one of its limits as a limit of an integral
• 9.2.2 Evaluate an integral that is improper because one of its limits is infinity
• 9.2.3 Rewrite an improper integral that has a domain restriction within its limits of integration as a sum of two integrals split at the discontinuity
• 9.2.4 Evaluate an integral that is improper because it has a discontinuity within its limits of integration
• 9.2.5 Recognize if an integral diverges or converges
• 9.2.6 Know the divergence or convergence of ∫(1/xp)dx
• 9.2.7 Be aware of unusual and interesting 2D shapes and 3D solids that involve infinite forms such as Gabriel's Horn including their geometric properties

Terminology

Terms you should be able to define: improper integral, diverges (divergent integral), converges (convergent integral), Gabriel's Horn

Mini-Lectures and Examples

Supplemental Resources (recommended)

Watch this classic math video Infinite Acres, MAA (1967) which provides a humorous take on the Gabriel's Horn paradox.

Supplemental Resources (optional)

rev. 2021-02-27