Volumes of Solids

Mean Value Theorem for Definite Integrals

Definition: Mean Value Theorem for Definite Integrals

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

f avg = 1 b a a b f ( x )   dx = f ( c )

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 ( x ) , that you find using the Average Value formula, 1 b a a b f ( x )   dx , has to equal the actual function’s y-value , f ( c ) , 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 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 avg = f ( c ) and it has to be exist in the x-interval given, [ a , b ].

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