# 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

 $\int_{a}^{b}\!{f(x)}\,\mathrm{d}{x}=f(\zeta)(b-a).$
###### Proof.

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

 $\displaystyle\int_{a}^{b}\!{f(x)}\,\mathrm{d}{x}$ $\displaystyle=\int_{a}^{b}\!{f(x)g(x)}\,\mathrm{d}{x}$ $\displaystyle=f(\zeta)\int_{a}^{b}\!{g(x)}\,\mathrm{d}{x}$ $\displaystyle=f(\zeta)\int_{a}^{b}\!{1}\,\mathrm{d}{x}$ $\displaystyle=f(\zeta)(b-a)$

as required. ∎

Title example of integral mean value theorem ExampleOfIntegralMeanValueTheorem 2013-03-22 18:20:24 2013-03-22 18:20:24 me_and (17092) me_and (17092) 4 me_and (17092) Example msc 26A06