Theorem 7.2.8: Mean Value Theorem for Integration
If f and g are continuous functions defined on
[a, b] so that g(x) 0,
then there exists a number c [a, b]
with
f(x) g(x) dx = f(c) g(x) dx
Proof
Define the numbersm = inf{ f(x): x [a, b] }Then we have m f(x) M and since g is non-negative we also have
M = sup{ f(x): x [a, b] }
m g(x) f(x) g(x) M g(x)By the properties of the Riemann integral this implies that
m g(x) dx f(x) g(x) dx M g(x) dxTherefore there exists a number d between m and M such that
d g(x) dx = f(x) g(x) dxBut since f is continuous on [a, b] and d is between m and M, we can apply the Intermediate Value Theorem to find a number c such that f(c) = d. Then
f(c) g(x) dx = f(x) g(x) dxwhich is what we wanted to prove.