Mean Value Theorem for Definite Integrals

Definition: Mean Value Theorem for Definite Integrals

Given a continuous function, $f\left(x\right)$,  the average y-value , $f_{\mathit{avg}}$, of that equation over a given closed x-interval , [ a , b ], will equal the function value, $f\left(c\right)$, at some x-value , c , inside the given x-interval , [ a , b ].

$f_{\mathit{avg}}=\frac{1}{b–a}\bullet \underset{a}{\overset{b}{\int }}f\left(x\right) \mathit{dx}\mathit{=}f\left(c\right)$

What it means:

The Mean Value Theorem for Definite Integrals is really just an extension of the Average Value formula.

What is being said is that the average y-value of an equation, $f\left(x\right)$, that you find using the Average Value formula, $\frac{1}{b–a}\bullet \underset{a}{\overset{b}{\int }}f\left(x\right) \mathit{dx}$, has to equal the actual function’s y-value , $f\left(c\right)$, and that the x-value where it happens, c , must happen inside the given x-interval , [ a , b ].

Basically, the average y-value has to exist, and it has to exist inside the given x-interval , [ a , b ].

Note: The Mean Value Theorem for Definite Integrals and the Average Value formula have similar sounding names, and similar looking equations. People some times think they are the same thing; they are ever so slightly different.

–           Average Value formula only gives you what the average y-value of an equation is, $f_{\mathit{avg}}$.

–           Mean Value Theorem for Definite Integrals tells you that the average y-value you found using the Average Value formula has to exist, $f_{\mathit{avg}}\mathit{=}f\left(c\right)$ and it has to be exist in the x-interval given, [ a , b ].