Applications of the Integral
Average Value of a Function
It is not entirely obvious what is meant by the average (or mean) value of a function on an interval. We know how to find the mean of a finite collection of numbers (their sum divided by their number). Needless to say, we run into problems when we want to talk about the mean of all the values of a function on a particular interval, since they are infinite in number.
To find our way out of this conundrum, we recall the definition of the n -th (upper) Riemann sum for the function f on the interval [a, b] :
|U n(f, a, b) = M i|
Note that U n(f, a, b) is equal to to the product of b - a (the length of the interval) and the mean of the values of f at n more or less evenly-spaced points in the interval. Clearly this is a reasonable approximate mean of the function f on the interval [a, b] .
Naturally, the same is true for the n th lower Riemann sum. As n gets larger and larger, we might imagine the upper and lower Riemann sums to approach (one from above, one from below) the product of b - a and some "true" mean of the function f on [a, b] . Indeed, this indicates precisely how we will define the average value, denoted . We set
|=||U n(f, a, b)|
|=||L n(f, a, b)|
There is a way of seeing graphically that this definition makes sense. An easy computation shows that the integral of the constant from a to b is equal to that of the function f (x) :
|=||(b - a)|
Thus, is the height of a rectangle of length b - a that will have the same area as the region below the graph of f (x) from a to b . In physical terms, if f (t) represents the velocity of a moving object, then another object moving with velocity will travel the same distance between the moments t = a and t = b .