Mean Value Theorem for Integrals

Theorem
Let $f$ be a continuous real function on the closed interval $\closedint a b$.

Then there exists a real number $k \in \closedint a b$ such that:


 * $\ds \int_a^b \map f x \rd x = \map f k \paren {b - a}$

Also see

 * Definition:Average Value of Function


 * Upper and Lower Bounds of Integral


 * Mean Value Theorem