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


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.


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


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

Mini-Lectures and Examples

STUDY: Calculus of Infinity - Improper Integrals

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)

Video: Improper Integrals, Selwyn Hollis's Video Calculus

Lesson: Improper Integrals, Dale Hoffman's Contemporary Calculus

rev. 2021-02-27