(problem 1a) Find the average value of the function over the interval .
The average value is .
We find the average value of a function.
Average value
We will define and compute the average value of a function on an interval.
example 1 Find the average value of over the interval .
Using the definition of average value, we have
Using the definition of average value, we have
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,
According to the definition of average value, we have To integrate the function , we need the trigonometric identity: Returning to our calculation,
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
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:
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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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