Answer
See explanation.
Work Step by Step
$ f $ is a continuous function with a closed interval $[a, b].$ Let $F(x)$ be a derivative of $ f $. By applying the mean-value theory to the $ F $, we get: there is a value of $x^{*}$ in $[a, b]$ that:
$f\left(x^{*}\right)=F^{\prime}\left(x^{*}\right)=\frac{-F(a)+F(b)}{-a+b}$
\[
\Rightarrow f\left(x^{*}\right)(-a+b)=-F(a)+F(b)=\int_{a}^{b} f(x) d x
\]
So we have proven the Mean-Value Theorem for Integrals.
