Average+Value+or+Mean+Value+Theorem+for+Integrals

=__Average Values and the Mean Value Theorem for Definite Integrals__=

//Finding Averages using Integration//
By now, you should be familiar with the process of finding the average value of a set of data. However, we cannot apply this directly to functions as the number of data points is at infinity. To average a function over a certain interval, we will use properties of integration:

The number of Chemistry labs completed per week by an AP Chemistry class is given by the equation: R(t)=1+(t) (1/2). Where t is the number of school weeks since school started. By the end of the first semester (18 weeks), how many labs per week did the class average?
 * Example One **

Solution:

Mean Value Theorem for Definite Integrals
Do you remember the rectangle approximations from Riemann Sums? When you used these approximations for areas under curves, you often found that the left or right-sided rectangles were an underestimate, that the other type was an overestimate, and that the Midpoint rectangles were much closer to the actual area.

Now we will expand on this idea. At some point between the far left side approximation and the far right side approximation, there will be a value of x such that the rectangle's area equals the actuals areas exactly. This is written mathematically as the Mean Value Theorem for Definite Integrals: In the rearranged form on the right, you will notice that at some point //c//, the value of the function equals the average value of the function.

Find a value //c// that satisfies the Mean Value Theorem for Integrals for the function f(x)=x 2 +2x+1 on the interval [-1,4].
 * Example Two **

Solution: The polynomial function is continuous, so we can apply the Mean Value Theorem: