San José State University

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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=∫abg(x)dx is positive. Then there exists an x* in (a, b) such that

∫abf(x)g(x)dx = f(x*)∫abg(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 ∫abg(x)dx. Then

mG ≤ ∫abf(x)g(x)dx ≤ MG
and hence
m ≤ (1/G)∫abf(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

f(x*) = (1/G)∫abf(x)g(x)dx

Therefore there exists an x* in (a, b) such that

∫abf(x)g(x)dx = f(x*)∫abg(x)dx

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