Classroom capsules: A generalization of the mean value theorem for integrals

Let f (x) be a continuous function on [a, b]. The Mean Value Theorem for Integrals asserts that there is a point c in (a, b) such that f^sup b^^sub a^ f (x) dx = f (c)(b - a). Unlike the proof of the Mean Value Theorem for derivatives, the proof of the Mean Value Theorem for Integrals typically does...

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Bibliographic Details
Published in:The College mathematics journal 2002-11, Vol.33 (5), p.408
Main Author: Tong, Jingcheng
Format: Article
Language:English
Subjects:
Algebra
Generalization
Geometry
Mathematics education
Theory
Online Access:Get full text
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Summary:Let f (x) be a continuous function on [a, b]. The Mean Value Theorem for Integrals asserts that there is a point c in (a, b) such that f^sup b^^sub a^ f (x) dx = f (c)(b - a). Unlike the proof of the Mean Value Theorem for derivatives, the proof of the Mean Value Theorem for Integrals typically does not use Rolle's Theorem. In this note, we use Rolle's Theorem to introduce a generalization of the Mean Value Theorem for Integrals. Our generalization involves two functions instead of one, and has a very clear geometric explanation.
ISSN:0746-8342
1931-1346