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 nth 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)
=
f (x)dx
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):
dx
=
|_{a}^{b}
=
(b - a)
=
f (x)dx
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.