The Mean Value Theorem for Integrals

Calculating average value of function over interval | AP Calculus AB | Khan Academy
Calculating average value of function over interval | AP Calculus AB | Khan Academy

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 [latex]f(x)[/latex] is continuous, a point [latex]c[/latex] exists in an interval [latex]\left[a,b\right][/latex] such that the value of the function at [latex]c[/latex] is equal to the average value of [latex]f(x)[/latex] over [latex]\left[a,b\right].[/latex] 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 [latex]f(x)[/latex] is continuous over an interval [latex]\left[a,b\right],[/latex] then there is at least one point [latex]c\in \left[a,b\right][/latex] such that

This formula can also be stated as

Proof

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

Dividing by [latex]b-a[/latex] gives us

Since [latex]\frac{1}{b-a}{\displaystyle\int }_{a}^{b}f(x)dx[/latex] is a number between [latex]m[/latex] and M, and since [latex]f(x)[/latex] is continuous and assumes the values [latex]m[/latex] and M over [latex]\left[a,b\right],[/latex] by the Intermediate Value Theorem (see Continuity), there is a number [latex]c[/latex] over [latex]\left[a,b\right][/latex] such that

and the proof is complete.

[latex]_\blacksquare[/latex]

Example: Finding the Average Value of a Function

Find the average value of the function [latex]f(x)=8-2x[/latex] over the interval [latex]\left[0,4\right][/latex] and find [latex]c[/latex] such that [latex]f(c)[/latex] equals the average value of the function over [latex]\left[0,4\right].[/latex]

We can see in Figure 1 that the function represents a straight line and forms a right triangle bounded by the [latex]x[/latex]– and [latex]y[/latex]-axes. The area of the triangle is [latex]A=\frac{1}{2}(\text{base})(\text{height}).[/latex] We have

The average value is found by multiplying the area by [latex]\frac{1}{(4-0)}.[/latex] Thus, the average value of the function is

Set the average value equal to [latex]f(c)[/latex] and solve for [latex]c[/latex].

At [latex]c=2,f(2)=4.[/latex]

The graph of a decreasing line f(x) = 8 – 2x over [-1,4.5]. The line y=4 is drawn over [0,4], which intersects with the line at (2,4). A line is drawn down from (2,4) to the x-axis and from (4,4) to the y axis. The area under y=4 is shaded.

Figure 1. By the Mean Value Theorem, the continuous function [latex]f(x)[/latex] takes on its average value at c at least once over a closed interval.

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

You can view the transcript for this segmented clip of “5.3 The Fundamental Theorem of Calculus” here (opens in new window).

Try It

Find the average value of the function [latex]f(x)=\dfrac{x}{2}[/latex] over the interval [latex]\left[0,6\right][/latex] and find [latex]c[/latex] such that [latex]f(c)[/latex] equals the average value of the function over [latex]\left[0,6\right].[/latex]

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

Given [latex]{\displaystyle\int }_{0}^{3}{x}^{2}dx=9,[/latex] find [latex]c[/latex] such that [latex]f(c)[/latex] equals the average value of [latex]f(x)={x}^{2}[/latex] over [latex]\left[0,3\right].[/latex]

We are looking for the value of [latex]c[/latex] such that

Replacing [latex]f(c)[/latex] with [latex]c[/latex]2, we have

Since [latex]\text{−}\sqrt{3}[/latex] is outside the interval, take only the positive value. Thus, [latex]c=\sqrt{3}[/latex] (Figure 2).

A graph of the parabola f(x) = x^2 over [-2, 3]. The area under the curve and above the x-axis is shaded, and the point (sqrt(3), 3) is marked.

Figure 2. Over the interval [latex]\left[0,3\right],[/latex] the function [latex]f(x)={x}^{2}[/latex] takes on its average value at [latex]c=\sqrt{3}.[/latex]

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

You can view the transcript for this segmented clip of “5.3 The Fundamental Theorem of Calculus” here (opens in new window).

Try It

Given [latex]{\displaystyle\int }_{0}^{3}(2{x}^{2}-1)dx=15,[/latex] find [latex]c[/latex] such that [latex]f(c)[/latex] equals the average value of [latex]f(x)=2{x}^{2}-1[/latex] over [latex]\left[0,3\right].[/latex]

[latex]c=\sqrt{3}[/latex]

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