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

h ( x ) = ( x − b ) ∫ a x f ( t ) d t + ( x − a ) ∫ x b g ( t ) d t .

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

∫ a c f ( t ) d t + ∫ c b g ( t ) d t = 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

∫ a c f ( t ) d t = 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)