Average Value of a Function

Let’s get started: You already know how to take the average of some finite set. The "mean" is the "average" you're used to, where you add up all the numbers and then divide by the number of numbers. Average = x = x 1 + x 2 + x 3 + …. + x n n

What if you need to find the average over infinitely many values (those that the function takes on over some interval) Must use CALCULUS! Before I show how to do this…let’s talk about WHY we might want to do this?

Consider the following picture: How high would the water level be if the waves all settled?

Okay! So, now that you have seen that this an interesting question... Let’s forget about real life, and... Do Some Math

Suppose we have a “nice”function and we need to find its average value over the interval [a,b].

Let’s apply our knowledge of how to find the average over a finite set of values to this problem: First, we partition the interval [a,b] into n subintervals of equal length to get back to the finite situation: In the above graph, we have n=8

Let us set up our notation: comes from the i-th interval Now we can get an estimate for the average value:

Let’s try to clean this up a little: Since

In a more condensed form, we now get: But we want to get out of the finite, and into the infinite! How do we do this? Take Limits!!!

In this way, we get the average value of f(x) over the interval [a,b]: So, if f is a “nice” function (i.e. we can compute its integral) then we have a precise solution to our problem.

Let’s look back at our graph:

So we’ve solved our problem! If I give you the equation and ask you to find it’s average value over the interval [0,2], you’ll all say NO PROBLEM!!!! the answer is…

Now we can answer our fish tank question! (That is, if the waves were described by an integrable function)