The Mean Value Theorem for Integrals

Learning Outcomes

  • Describe the meaning of the Mean Value Theorem for Integrals

The Mean Value Theorem for Integrals states that a continuous function on a closed interval takes on its average value at the same point in that interval. The theorem guarantees that if f(x) is continuous, a point c exists in an interval [a,b] such that the value of the function at c is equal to the average value of f(x) over [a,b]. We state this theorem mathematically with the help of the formula for the average value of a function that we presented at the end of the preceding section.

The Mean Value Theorem for Integrals


If f(x) is continuous over an interval [a,b], then there is at least one point c[a,b] such that

f(c)=1baabf(x)dx.

 

This formula can also be stated as

abf(x)dx=f(c)(ba).

 

Proof

Since f(x) is continuous on [a,b], by the extreme value theorem (see Maxima and Minima), it assumes minimum and maximum values—m and M, respectively—on [a,b]. Then, for all x in [a,b], we have mf(x)M. Therefore, by the comparison theorem (see The Definite Integral), we have

m(ba)abf(x)dxM(ba).

 

Dividing by ba gives us

m1baabf(x)dxM.

 

Since 1baabf(x)dx is a number between m and M, and since f(x) is continuous and assumes the values m and M over [a,b], by the Intermediate Value Theorem (see Continuity), there is a number c over [a,b] such that

f(c)=1baabf(x)dx,

 

and the proof is complete.

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Example: Finding the Average Value of a Function

Find the average value of the function f(x)=82x over the interval [0,4] and find c such that f(c) equals the average value of the function over [0,4].

Watch the following video to see the worked solution to Example: Finding the Average Value of a Function.

Try It

Find the average value of the function f(x)=x2 over the interval [0,6] and find c such that f(c) equals the average value of the function over [0,6].

Hint

Use the procedures from the last example to solve the problem

example: FINDING THE POINT WHERE A FUNCTION TAKES ON ITS AVERAGE VALUE

Given 03x2dx=9, find c such that f(c) equals the average value of f(x)=x2 over [0,3].

Watch the following video to see the worked solution to Example: Finding the Point Where a Function Takes on its Average Value.

Try It

Given 03(2x21)dx=15, find c such that f(c) equals the average value of f(x)=2x21 over [0,3].