Mean Value Theorem for Integrals

Mean Value Theorem for Integrals
Let $f$ be a continuous real function on the closed interval $\left[{a..b}\right]$.

Then there exists a real number $k$ in the interval such that:


 * $\displaystyle \int_a^b f \left({x}\right) \ \mathrm d x = f \left({k}\right) \left({b - a}\right)$

Proof
From Continuous Function is Riemann Integrable, $f$ is Riemann integrable on $\left[{a .. b}\right]$.

By the Extreme Value Theorem, there exist $m, M \in \left[{a .. b}\right]$ such that:


 * $\displaystyle f \left({m}\right) = \min_{x \in \left[{a .. b}\right]} f \left({x}\right)$
 * $\displaystyle f \left({M}\right) = \max_{x \in \left[{a .. b}\right]} f \left({x}\right)$

From Relative Sizes of Definite Integrals, it follows that:


 * $\displaystyle \int_a^b f \left({m}\right) \ \mathrm d x \leq \int_a^b f \left({x}\right) \ \mathrm d x

\leq \int_a^b f \left({M}\right) \ \mathrm d x$

Now, Integration of a Constant allows for computing these bounds, thus obtaining:


 * $\displaystyle f \left({m}\right) \left({b - a}\right) \leq \int_a^b f \left({x}\right) \ \mathrm d x \leq f \left({M}\right) \left({b - a}\right)$

Dividing all terms by $\left({b - a}\right)$ gives:


 * $\displaystyle f \left({m}\right) \leq \frac 1 {b-a}\int_a^b f \left({x}\right) \ \mathrm d x \leq f \left({M}\right)$

By the Intermediate Value Theorem, there exists some $k \in \left({a..b}\right)$ such that:

Also see

 * Average Value of a Function
 * Upper and Lower Bounds of Integral