Average+Value+of+a+Function+and+Mean+Value+Theorem+for+Integrals


 * __Average Value of a Function__**

If you are trying to find the average value of a function f(x), you'll need to take the definite integral on the interval [a,b] that you are looking for and divide by the difference between the two(b-a).

For example:

if you are looking for the average value of over the interval [1,3], you'll need to integrate first.



Next, you take the definite integral and divide by the difference.

26/3/(2)=13/3

This value 13/3 is the average value of f(x) for the interval [1,3]. Because this is an average value, there is a value, x=c, for which f(x)=13/3. By setting, you can solve for the c value. In this instance, c= or In mathematical notation, it would look like this:



graphically, it would look like this: media type="youtube" key="OXDLZW2D0lo" height="385" width="640"

__**Integral Mean Value Theorem**__

The integral mean value theorem takes this a step further. It states for any function f(x) there is a point f(c) that is equal to the average of the function on the interval [a,b] given that it is continuous. Mathematically this would look like this:   again, f(c) is a point on the graph of f(x) on the interval [a,b]

this looks like this: media type="youtube" key="zF-19UmZGOc" height="385" width="480"  For this particular graph, c=3 and f(3) appears to be the average value of f(x) on the interval [0,6].

All of this is highlighted in the youtube video:

media type="youtube" key="Dud7NVdQ24g" height="385" width="480"

__**Solving for C**__

Example: For the Equation the average value over the interval [2,6] is 67/3 to find the c value for the function, you need to set this as the average as the y value.



Now solve:

The solutions for this equation are: and

Because the first solution is not on the interval, we know that c=

Usually in the problems you will encounter, you will need to find this c value, so it is important that you understand how to apply the average value of a function to find this c value. Hopefully this helps your understanding of average values of a function and the integral mean value theorem.