2 Let’s get started:You already know about functions and how to take the average of some finite set.Today we’re going to take the average over infinitely many values (those that the function takes on over some interval)…which means CALCULUS!Before I show how to do this…let’s talk about WHY we might want to do this?
3 Consider the following picture: How high would the water level be if the waves all settled?
4 Okay! So, now that you have seen that this an interesting question... Let’s forget about real life, and...Do Some Math
5 Suppose we have a “nice”function and we need to find its average value over the interval [a,b].
6 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 ofequal length to get back to the finite situation:In the above graph, we have n=8
7 Let us set up our notation: comes from the i-th intervalNow we can get an estimate for theaverage value: