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An Extended Mean Value Theorem for Integrals

Let f(x) be a continuous function on the interval [a, b] and g(x) a non-negative integrable function over [a, b] such that G=∫_{a}^{b}g(x)dx
is positive.
Then there exists an x* in (a, b) such that

∫_{a}^{b}f(x)g(x)dx = f(x*)∫_{a}^{b}g(x)dx

Proof:

Let m be the greatest lower bound for f(x) on the interval [a, b] and M be the lowest upper bound for f(x) on that same interval. Let G be the value of
∫_{a}^{b}g(x)dx. Then

mG ≤ ∫_{a}^{b}f(x)g(x)dx ≤ MG
and hence
m ≤ (1/G)∫_{a}^{b}f(x)g(x)dx ≤ M

Since f(x) is a continuous function on [a, b] it attains all values between m and M. This means there is an x* in (a, b) such that