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.