Volumes of Solids

Extension of Mean Value Theorem for Definite Integrals

Cool Extension:

Very often AP calc teachers like to extend the Mean Value Theorem for Definite Integrals to show off a cool connection.

If you take the original Mean Value Theorem for Definite Integrals formula:

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

And then do one little move of algebra, multiply the entire equation by ( b a ), to get the definite integral alone in the middle of the equals.

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

You then simplify the algebra work by canceling out the ( b a ) term in the middle, and get the conclusion:

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

Meaning: What this is saying is the exact net area between a curve, f ( x ) , and the x-axis , a b f ( x )   dx , is equal to the area of the rectangle (think Riemann Sums, base x height ), ( b a ) f ( c ) .

a b f ( x )   dx

=

( b a ) f ( c )

 

=

The area of that little rectangle on the right is equal to the net area on the left. Remember that net area means adding the positive and the negative areas to get the final result.

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