A Generalization of the Mean Value Theorem for Integrals

Abstract

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 fab 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.