Section 6.5: The Average Value of a Function
1. Defining the Average Value of a Function
Suppose that y1,...,ynare a set of numbers. In order to find the
average value of these numbers, we sum them up and divide by n. That
is, Average Value= y1+...yn
n. This is useful for discrete mathematics, but
usually in mathematics we deal with continuous models (else we cannot
apply Calculus). How should we try to find averages for continuous
models? Consider the following example.
Example 1.1. Suppose that the temperature C(t) is a function of time
in hours, 1 6t624. In order to calculate the average temperature
over a 24 hour period, we could measure the temperature every hour
and then take the average of these measurements:
Average Value ∼C(1) + C(2) + ···+C(24)
24 .
However, this does not take into account that the temperature could
suddenly spike or drop at any point. Therefore, to find a better ap-
proximation, we should take smaller time intervals. In general, on the
interval [0,24], we could break it up into nintervals, each of length
∆x= (b−1)/n = 24/n. Then we approximate using a time value in
each interval and get:
Average ≃C(x1) + ···+C(xn)
n.
But we know ∆x= 24/n, so we get
Average ≃C(x1) + ···+C(xn)
n=C(x1) + ···+C(xn)
24/∆x
=(C(x1) + ···+C(xn))∆x
n=1
24
n
X
i=1
C(xi)∆x.
This last sum is a Riemann sum, and as we take smaller and smaller
subdivisions, we get closer and closer to the definite integral. Thus we
could define:
Average ≃lim
n→∞
1
b−a
n
X
i=1
C(xi)∆x=1
24 Zb
a
C(x)dx.
What we did for this example is not unique to this example. It works in
precisely the same way for any continuous function. The only difference
we note is that if we are considering the average of a general interval
[a, b], we get ∆x= (b−a)/n. In general, we have:
1
