On-line Math 21

6.2  Average value of a function

The concept of an average is easy. To find the average (or mean) score on an exam, we simply add up all the scores and divide by the number of people who took the exam. More generally, the average of a collection {y₁,¼,y_n} of n numbers is

yav: = y1+y2+¼+yn n .

It is perhaps a little less clear what we would mean by the average value of a function defined over an interval, but the concept is really the same. To find the average of f(x) over the interval [a,b] , you could simply find a bunch of points in there, add up the values of the function on all those points, and divide by the total number of points you chose. But, of course this is only approximately the average, since you just chose a few points. To make it more accurate, you'd just take more points, and in the end, you'd have meaningless infinities top and bottom.

This is the first application of integration where the integral is not quite so apparently an integral. In this case, if, say you look at values of f(x) at {x₁,¼,x_n} , {f(x₁),¼,f(x_n)} , then the numerator of the average is (f(x₁)+¼+f(x_n)) , which is not an integral, even approximately. But if we multiply each of those terms by dx , it does look more like an integral formula, (f(x₁)dx+¼+f(x_n)dx) . Remember that dx is just the small distance in the x direction, the width of the subinterval we're at. But, of course we have to multiply the bottom (n) by dx as well, which at first doesn't make any sense, n dx (at least not in the limit as dx® 0 and n® ¥). But, if you write n = 1+¼+1 ( n times, of course), then n dx = dx+dx+¼+dx , which looks, now, like the integral of 1 .

So, the average value of f(x) , for x Î [a,b] , is

fav: = ó õ b a  f(x) dx ó õ b a  1 dx = 1 b-a ó õ b a  f(x) dx,

which is the limit of these sums as dx® 0 . Of course,

ó õ b a  1 dx = b-a.

As with many of these applications, the examples are not the point. The thing to concentrate on is the derivation of the formula. However, some examples will help you with the homework.

Example 1 Find the average value of sin(x) over

1. [0,p] ,
2. [0,2p]

Solution

Exercise 1 Find the average value of f(x): = sin²(x) on the interval [0,p] . f_av =

Exercise 2 Egbert is slicing oranges. He slices them by setting them on a table, and giving them a good thwack with a very sharp knife. Unfortunately, he's doesn't have very good aim, and his thwack could, with equal likelihood, hit the orange anywhere. What is the average size of the cut surface of Egbert's oranges? He does always at least nick the oranges (when he doesn't, he doesn't count that wiff as a thwack), and the oranges are all perfect spheres of radius 2 inches.

Answer:

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