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Mathematical Expression Editor

We find the average value of a function.

1 Average value

We will define and compute the average value of a function on an interval.

Average Value The average value of an integrable function, , on the interval is given
by

example 1 Find the average value of over the interval . Using the definition of average value, we have

(problem 1a) Find the average value of the function over the interval . The average value is .

(problem 1b) Find the average value of the function over the interval . The average value is .

(problem 1c) Find the average value of the function over the interval . The average value is .

(problem 1d) Find the average value of the function over the interval . The average value is .

example 2 Find the average value of on the interval . According to the definition of average value, we have To integrate the function , we
need the trigonometric identity: Returning to our calculation,

(problem 2) Find the average value of over the interval .

use the trigonometric identity

The average value is

example 3 Find the average value of over the interval . Using the definition of average value, we have We now use a -substitution, with
and , thus

Continuing our average value calculation, we have

(problem 3) Find the average value of over the interval . First, use a u-substitution to compute the indefinite integral: Now find the average
value of the function:

Mean Value Theorem for Integrals If is continuous on the interval then there is
a value between and such that , i.e., for some number in the interval
.

For a positive function, , the Mean Value theorem for Integrals implies that the area
under the graph of over the interval is equal to the area of a rectangle with base
and height .

(problem 4a) In problem 1a, we found that the average value of on the interval was
3. Find the value(s) of from the Mean Value Theorem for Integrals for this
situation.

(problem 4b) In problem 1b, we found that the average value of on the interval was .
Find the value(s) of from the Mean Value Theorem for Integrals for this
situation.

(problem 4c) In problem 1c, we found that the average value of on the interval was .
Find the value(s) of from the Mean Value Theorem for Integrals for this
situation.

Here is a detailed, lecture style video on average value:

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2 Theoretical Justifications

In this section, we derive the formula for average value. Recall that we are trying to
find the average value of a function, , over an interval, . We begin by subdividing the
interval into equal sub-intervals, each of length In each of these sub-intervals, we
choose a sample point. We denote the sample point in the -th sub-interval by . As an
approximation to the average value of the function over the interval we take the
average of the function values at the sample points: This can be written in
summation notation as Observe that equation in the definition of can be
rewritten as This allows us to rewrite our approximation of as Since is
a constant, we can bring it inside the summation to write: Finally, the
approximation improves as the number of sample points, , increases. Therefore, we
define which by the definition of the definite integral can be written as