Mean Value Theorem for integrals

Mean Value Theorem for integrals: Let $f:[a,b]\to\mathbb{R}$ be continuous . Then there exists $c\in[a,b]$ such that $\int_a^b f(x)\,dx = f(c)\,(b-a).$ Equivalently, $f(c)=\frac{1}{b-a}\int_a^b f(x)\,dx.$

This result formalizes the idea that a continuous function takes its average value somewhere. It is frequently used to prove existence statements and to estimate integrals.

Proof sketch: By continuity on $[a,b]$, $f$ attains a minimum $m$ and maximum $M$. Then $m(b-a)\le \int_a^b f(x)\,dx \le M(b-a).$ Divide by $b-a$ to place the average value between $m$ and $M$, and apply the intermediate value theorem to $f$.