example of integral mean value theorem

Example.

If $f$ is a continuous real function on an interval $[a,b]$, then there exists a $\zeta \in (a,b)$ such that

$${\int }_a^{b}f(x)dx=f(\zeta )(b-a).$$

Proof.

Let $g(x)\equiv 1$. Then by the Integral Mean Value Theorem, there exists $\zeta \in (a,b)$ such that

${\displaystyle {\int }_a^b}f(x)dx$$={\displaystyle {\int }_a^b}f(x)g(x)dx$

$=f(\zeta ){\displaystyle {\int }_a^b}g(x)dx$

$=f(\zeta ){\displaystyle {\int }_a^b}1dx$

$=f(\zeta )(b-a)$

as required.∎