Chapter 5.4, Problem 69E

Generalizing the Mean Value Theorem for Integrals Suppose f and g are continuous on [a, b] and let

$h(x)=(x-b){\displaystyle {\int }_a^{x}f(t)dt+(x-a)}{\displaystyle {\int }_x^{b}g(t)dt.}$

1. a. Use Rolle’s Theorem to show that there is a number c in (a, b) such that

${\displaystyle {\int }_a^{c}f(t)dt}+{\displaystyle {\int }_c^{b}g(t)dt=f(c)(b-c)+g(c)(c-a),}$

which is a generalization of the Mean Value Theorem for Integrals.

1. b. Show that there is a number c in (a, b) such that

${\displaystyle {\int }_a^{c}f(t)dt=f(c)(b-c).}$

1. c. Use a sketch to interpret part (b) geometrically.
2. d. Use the result of part (a) to give an alternative proof of the Mean Value Theorem for Integrals.

(Source: The College Mathematics Journal, 33, 5, Nov 2002)