Generalizing the Mean Value Theorem for Integrals Suppose f and g are continuous on [a, b] and let h(x) = = (x b) a. Use Rolle's Theorem to show that there is a number c in (a, b) such that a ["* f(t) dt + (x a) [*g(t). dt. f(t) dt + t + f g ( g(t) dt = f(c)(b-c) + g(c) (c-a), which is a generalization of the Mean Value Theorem for Integrals. b. Show that there is a number c in (a, b) such that ff(t) dt = f(c)(b c). c. Use a sketch to interpret part (b) geometrically. d. Use the result of part (a) to give an alternative proof of the Mean Value Theorem for Integrals.