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Example Questions
Example Question #141 : Integral Applications
What is the average value of the function  on the intervalÂ
 ?
The average value of the function  on the intervalÂ
 isÂ
.
We can integrate the indefinite integralÂ
Â
as follows:
Set ,Â
.
Then Â
 orÂ
andÂ
.
Therefore,Â
Therefore, the average value is:
Example Question #2062 : Calculus Ii
What is the average value of the function  on the intervalÂ
 ?
The average value of the function  on the intervalÂ
 is equal toÂ
.
Substitute . Subsequently,Â
 andÂ
. The bounds of integration becomeÂ
 andÂ
, so the integral can be rewritten as
.
Example Question #1 : Average Values And Lengths Of Functions
What is the arc length if ,Â
 fromÂ
?
Write the formula for arc length.
Calculate the derivatives.
Substitute the derivatives and the bounds into the integral.
Example Question #2 : Average Values And Lengths Of Functions
What is the average value of the function
 Â
from  to
?Â
The average value of a function p(t) from t=a to t=b is found with the integral
.Â
In this case, we must compute the value of the integral
.
A substitution makes this integral clearer. Let . ThenÂ
. We should also rewrite the limits of integration in terms of u. When t = 0, u=1, and when t = 2, u = 5. Making these substitutions results in the integral
Evaluating this integral using the fact that
 yields
Example Question #3 : Average Values And Lengths Of Functions
What is the average value of the function  over the intervalÂ
?
In general, the average value of a function  over the intervalÂ
 isÂ
Â
This means that the average value of  overÂ
 isÂ
.
Since the antiderivative of  isÂ
, the integral evaluates toÂ
.
Example Question #1 : Average Values And Lengths Of Functions
What is the length of the curve  over the intervalÂ
?
The general formula for finding the length of a curve  over an intervalÂ
 isÂ
In this example, the arc length can be found by computing the integral
.
The derivative of  can be found using the power rule,Â
, which leads toÂ
.Â
At this point, a substitution is useful.
LetÂ
.
We can also express the limits of integration in terms of  to simplify computation. WhenÂ
, and whenÂ
.
Making these substitutions leads toÂ
.
Now use the power rule, which in general is , to evaluate the integral.Â
Example Question #1 : Average Values And Lengths Of Functions
Given the interval, find the average value of the following function:
When f is integrable on [a,b], the average value of f(x) on [a,b] is defined to be:
For the problem statement, we are given f(x) and the intervals [a,b]. All that needs to be done is solving the integral over this interval and dividing the result by the difference between the two intervals.Â
So:
To solve this integral, we have two options. We can FOIL the terms out for  and solve the integral of the resulting polynomial, or we can use a simple u-substitution. Either way, the result will always be the same. We will try both ways, to prove that this holds true:
FOIL Method:
This is one of the answer choices!
U-Substition Method:
Next, we must adjust the bounds of the new integral which will be in terms of u.
So the new integral becomes:
As you can see both methods results in the same answer!
Â
Example Question #6 : Average Values And Lengths Of Functions
What is the average of value of  between the intervalsÂ
 andÂ
 ?Â
When asked for the average value between intervals of form  andÂ
 then integral becomesÂ
 so in this case we can rewrite our average value problem asÂ
.
The integral comes  evaluated atÂ
 andÂ
 which simplifies toÂ
 (note thatÂ
 is 1).
However, since we are looking for the average value, we must divide the whole thing by b-a, which in this case is 2, so the final answer isÂ
.
Example Question #7 : Average Values And Lengths Of Functions
What is the average value of the function  on the intervalÂ
?
The average value of a function over an interval is defined to be the integral of the function divided by the length of the interval.
The general rule for this type of integration is as follows.
So we get:
Example Question #8 : Average Values And Lengths Of Functions
What is the average value of the function
on the interval, ?
The average value of a function is found by taking the integral of the function over the interval and dividing by the length of the interval.
The general rule for this type of integration is as follows.
So we have:
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