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The Average Value of a Continuous Function on a Closed Interval

Author: Sophia
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what's covered
In this lesson, you will learn about the average value of a continuous function over an interval open square brackets a comma space b close square brackets. Recall that the average of a set of numbers is the sum of the numbers, divided by the number of numbers. This takes on a different meaning for continuous functions. Specifically, this lesson will cover:

Table of Contents

1. The Idea Behind Average Value

When finding the average of a set of numbers, you add up all the numbers, then divide by how many numbers there are.

EXAMPLE

Given the numbers 81, 85, 89, and 71, the average of these four numbers is fraction numerator 81 plus 85 plus 89 plus 71 over denominator 4 end fraction equals 81.5.

Now consider a function y equals f open parentheses x close parentheses on some interval open square brackets a comma space b close square brackets. Break up the interval open square brackets a comma space b close square brackets into n equal subintervals. Then, select a value of x from each subinterval. Call these values x subscript 1 comma space x subscript 2 comma space horizontal ellipsis comma space x subscript n.

Then, the average of these values is fraction numerator f open parentheses x subscript 1 close parentheses plus f open parentheses x subscript 2 close parentheses plus horizontal ellipsis plus f open parentheses x subscript n close parentheses over denominator n end fraction equals sum from k equals 1 to n of open square brackets f open parentheses x subscript k close parentheses times 1 over n close square brackets.

The summation resembles a Riemann sum, but the increment x term is missing inside the summation. Recall that increment x equals fraction numerator b minus a over denominator n end fraction.

We can multiply the summation by fraction numerator b minus a over denominator b minus a end fraction as follows:

sum from k equals 1 to n of open square brackets f open parentheses x subscript k close parentheses times fraction numerator b minus a over denominator b minus a end fraction 1 over n close square brackets
equals sum from k equals 1 to n of open square brackets f open parentheses x subscript k close parentheses times fraction numerator b minus a over denominator n end fraction fraction numerator 1 over denominator b minus a end fraction close square brackets

We replace fraction numerator b minus a over denominator n end fraction with increment x colon

equals sum from k equals 1 to n of open square brackets f open parentheses x subscript k close parentheses times increment x times fraction numerator 1 over denominator b minus a end fraction close square brackets

Since fraction numerator 1 over denominator b minus a end fraction is a constant, it can be factored out and written in front of the summation:

fraction numerator 1 over denominator b minus a end fraction sum from k equals 1 to n of open square brackets f open parentheses x subscript k close parentheses times increment x close square brackets

Recall that the summation sum from k equals 1 to n of open square brackets f open parentheses x subscript k close parentheses times increment x close square brackets approaches the value of integral subscript a superscript b f open parentheses x close parentheses d x as n rightwards arrow infinity as long as f open parentheses x close parentheses is integrable on open square brackets a comma space b close square brackets. Since we are assuming f open parentheses x close parentheses is continuous on open square brackets a comma space b close square brackets comma f open parentheses x close parentheses is also integrable on open square brackets a comma space b close square brackets.

Note that the summation for the average value is the Riemann sum for f open parentheses x close parentheses but multiplied by fraction numerator 1 over denominator b minus a end fraction.

This leads to an integral formula to find the average value of a continuous function f open parentheses x close parentheses on an interval open square brackets a comma space b close square brackets.

formula to know
Average Value of a Function
If f open parentheses x close parentheses is continuous on the closed interval open square brackets a comma space b close square brackets comma then the average value of f open parentheses x close parentheses on open square brackets a comma space b close square brackets is fraction numerator 1 over denominator b minus a end fraction integral subscript a superscript b f open parentheses x close parentheses d x.

big idea
For a geometric interpretation of average value, let H equals the average value of a nonnegative function f open parentheses x close parentheses on open square brackets a comma space b close square brackets. The figure below shows an illustration of this.

Two graphs labeled ‘(1)’ and ‘(2)’ with an x-axis and a y-axis each. The graph labeled ‘(1)’ consists of two dashed lines, which rise from the points labeled ‘a’ and ‘b’ on the x-axis such that b is greater than a. A curve, y equals f(x), descends from a point on the dashed line x equals a, reaches an inverted peak, and then rises to a point near the top of the dashed line x equals b, which is higher than the point on the dashed line x equals a. The area below the curve and above the x-axis is shaded from the dashed lines x equals a to x equals b. The graph labeled ‘(2)’ consists of two dashed lines, which rise from the points labeled ‘a’ and ‘b’ on the x-axis such that b is greater than a. A curve, y equals f(x), descends from a point on the dashed line x equals a, reaches an inverted peak, and then rises to a point near the top of the dashed line x equals b, which is higher than the point on the dashed line x equals a. A horizontal dashed line y equals H starts from a point labeled ‘H’ on the y-axis, intersecting the dashed lines x equals a and x equals b above the starting point of the curve and below the starting point of the curve, respectively. The area below the line y equals H and above the x-axis from the dashed lines x equals a to x equals b is shaded.

  • The graph in (1) is the region bounded by the graph of f open parentheses x close parentheses and the x-axis on open square brackets a comma space b close square brackets.
  • The graph in (2) is the rectangle with an area equal to integral subscript a superscript b f open parentheses x close parentheses d x. Note that the base is b minus a comma and its height is H, where H is the average value of f open parentheses x close parentheses on open square brackets a comma space b close square brackets.
The area of the rectangle with height H and width b minus a is equal to the area of the region bounded by the graph of f open parentheses x close parentheses and the x-axis on open square brackets a comma space b close square brackets.

2. Computing the Average Value of a Continuous Function

Now that we have a formula for average value, let’s compute and interpret average values.

EXAMPLE

Find the average value of f open parentheses x close parentheses equals sin x on the interval open square brackets 0 comma space straight pi close square brackets.

From the formula, this is equal to fraction numerator 1 over denominator straight pi minus 0 end fraction integral subscript 0 superscript straight pi sin x d x equals fraction numerator begin display style 1 end style over denominator begin display style straight pi end style end fraction integral subscript 0 superscript straight pi sin x d x.

Now, we evaluate the definite integral:

fraction numerator begin display style 1 end style over denominator begin display style straight pi end style end fraction integral subscript 0 superscript straight pi sin x d x Start with the original expression.
equals open 1 over straight pi open parentheses short dash cos x close parentheses close vertical bar subscript 0 superscript straight pi Apply the fundamental theorem of calculus.
equals 1 over straight pi open parentheses short dash cosπ close parentheses minus 1 over straight pi open parentheses short dash cos 0 close parentheses Substitute the upper and lower endpoints.
equals 1 over straight pi plus 1 over straight pi Evaluate the parentheses.
equals 2 over straight pi Simplify.

The average value of f open parentheses x close parentheses equals sin x on the interval open square brackets 0 comma space straight pi close square brackets is equal to 2 over straight pi.

To see the geometric interpretation, here is the graph of the region bounded by f open parentheses x close parentheses equals sin x and the x-axis on the interval open square brackets 0 comma space straight pi close square brackets and the rectangle whose height is the average value and whose width is straight pi. Note: 2 over straight pi almost equal to 0.64

A graph with an x-axis ranging from 0 to π and a y-axis ranging from 0 to 1. A curve, y equals sin x, rises from a point (0, 0), reaches a peak, and then descends to meet the x-axis at π. A dashed line extends horizontally from the marked point at (0, 0.64), intersecting the curve at two points. The area between the curve, y equals sin x, and the x-axis is shaded.

watch
Find the average value of f open parentheses x close parentheses equals fraction numerator 15 x over denominator x squared plus 1 end fraction on the interval open square brackets 0 comma space 5 close square brackets.

try it
Consider the function f open parentheses x close parentheses equals x squared plus 2.
Find the average value of f  (x  ) on the interval [0, 4]. 
The average value of f open parentheses x close parentheses equals x squared plus 2 on the interval open square brackets 0 comma space 4 close square brackets is found by calculating fraction numerator 1 over denominator 4 minus 0 end fraction integral subscript 0 superscript 4 open parentheses x squared plus 2 close parentheses d x equals 1 fourth integral subscript 0 superscript 4 open parentheses x squared plus 2 close parentheses d x.

Evaluate the integral:

open equals 1 fourth open parentheses 1 third x cubed minus 2 x close parentheses close vertical bar subscript 0 superscript 4 integral x squared d x equals 1 third x cubed integral 2 d x equals 2 x
equals 1 fourth open square brackets open parentheses 1 third open parentheses 4 close parentheses cubed minus 2 open parentheses 4 close parentheses close parentheses minus open parentheses 1 third open parentheses 0 close parentheses cubed minus 2 open parentheses 0 close parentheses close parentheses close square brackets Evaluate when x equals 4 and x equals 0 comma then subtract.
equals 1 fourth open square brackets open parentheses 88 over 3 close parentheses minus 0 close square brackets Simplify within each group of parentheses.
equals 22 over 3 Simplify.

The average value of f open parentheses x close parentheses equals x squared plus 2 on the interval open square brackets 0 comma space 4 close square brackets is equal to 22 over 3.

try it
Consider the function f open parentheses x close parentheses equals 4 over x squared.
Find the average value of f  (x  ) on the interval [1, 2]. 
The average value of f open parentheses x close parentheses equals 4 over x squared on the interval open square brackets 1 comma space 2 close square brackets is found by calculating fraction numerator 1 over denominator 2 minus 1 end fraction integral subscript 1 superscript 2 4 over x squared d x.

Evaluate the integral:

equals integral subscript 1 superscript 2 4 x to the power of short dash 2 end exponent d x fraction numerator 1 over denominator 2 minus 1 end fraction equals 1 over 1 equals 1 comma so there is no need to have a constant outside the integral.
Also, rewrite 4 over x squared as 4 x to the power of short dash 2 end exponent so that the power rule can be used. 
open equals short dash 4 x to the power of short dash 1 end exponent close vertical bar subscript 1 superscript 2 Find the antiderivative using the power rule.
integral 4 x to the power of short dash 2 end exponent d x equals 4 open parentheses fraction numerator 1 over denominator short dash 1 end fraction close parentheses x to the power of short dash 1 end exponent equals short dash 4 x to the power of short dash 1 end exponent
equals open parentheses short dash 4 open parentheses 2 close parentheses to the power of short dash 1 end exponent close parentheses minus open parentheses short dash 4 open parentheses 1 close parentheses to the power of short dash 1 end exponent close parentheses Substitute x equals 2 and x equals 1 comma then subtract.
equals open parentheses short dash 2 close parentheses minus open parentheses short dash 4 close parentheses Simplify within each group of parentheses.
2 to the power of short dash 1 end exponent equals 1 half and 1 to the power of short dash 1 end exponent equals 1.
equals 2 Simplify.

The average value of f open parentheses x close parentheses equals 4 over x squared on the interval open square brackets 1 comma space 2 close square brackets is equal to 2.

In each case, the units of f open parentheses x close parentheses and the units of the average value of f open parentheses x close parenthesesare the same. So if f open parentheses x close parentheses was measured in feet, then the average value would also be measured in feet.

EXAMPLE

During a 9-hour workday, the production rate at time t hours is r open parentheses t close parentheses equals 5 plus square root of t cars per hour. What is the average hourly production rate?

We seek the average value of r open parentheses t close parentheses over the interval open square brackets 0 comma space 9 close square brackets.

A v e r a g e space V a l u e equals fraction numerator 1 over denominator 9 minus 0 end fraction integral subscript 0 superscript 9 open parentheses 5 plus t to the power of 1 divided by 2 end exponent close parentheses d t Start with the original expression. Rewrite square root of t equals t to the power of 1 divided by 2 end exponent to be able to use the power rule.
equals 1 over 9 open open parentheses 5 t plus 2 over 3 t to the power of 3 divided by 2 end exponent close parentheses close vertical bar subscript 0 superscript 9 Apply the fundamental theorem of calculus.
equals 1 over 9 open square brackets 5 open parentheses 9 close parentheses plus 2 over 3 open parentheses 9 close parentheses to the power of 3 divided by 2 end exponent close square brackets minus 1 over 9 open square brackets 5 open parentheses 0 close parentheses plus 2 over 3 open parentheses 0 close parentheses to the power of 3 divided by 2 end exponent close square brackets Substitute the upper and lower endpoints.
equals 1 over 9 open parentheses 45 plus 18 close parentheses minus 1 over 9 open parentheses 0 close parentheses Evaluate.
equals 7 Simplify.

The average rate of production is 7 cars per hour.

Shown in the figure is the region between r open parentheses t close parentheses equals 5 plus square root of t and the t-axis, as well as the horizontal line r open parentheses t close parentheses equals 7. Note that the area between r open parentheses t close parentheses and the t-axis is equal to the area of the rectangle with the same base (9) and height 7 (the average value).

A graph with an x-axis ranging from 0 to 9 and a y-axis ranging from 0 to 8. The curve rises from the point (0, 5) and extends until the point (9, 8) by passing through the marked point at (4, 7). A horizontal line starts from the marked point at (0, 7) and continues until the point (10, 7) by passing through another marked point at (4, 7). The area below the curve up to the x-axis is shaded.

summary
In this lesson, you began by understanding the idea behind average value, following the path from the formula to find the average of a set of numbers to an integral formula to find the average value of a continuous function f open parentheses x close parentheses on an interval open square brackets a comma space b close square brackets. You also learned how the fundamental theorem of calculus can be used to compute the average value of a continuous function f open parentheses x close parentheses on an interval open square brackets a comma space b close square brackets.

Source: THIS TUTORIAL HAS BEEN ADAPTED FROM CHAPTER 4 OF "CONTEMPORARY CALCULUS" BY DALE HOFFMAN. ACCESS FOR FREE AT WWW.CONTEMPORARYCALCULUS.COM. LICENSE: CREATIVE COMMONS ATTRIBUTION 3.0 UNITED STATES.

Formulas to Know
Average Value of a Function

If f open parentheses x close parentheses is continuous on the closed interval open square brackets a comma space b close square brackets comma then the average value of f open parentheses x close parentheses on open square brackets a comma space b close square brackets is fraction numerator 1 over denominator b minus a end fraction integral subscript a superscript b f open parentheses x close parentheses d x.

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