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.
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
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):
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.
Mi
Un(f, a, b)
f (x)dx
