
In the story of Gauss’s problem of adding up the numbers from 1 to100, one interpretation of the result, 5,050, is that the average of all the numbers from 1 to 100 is 50.5. This is the ordinary definition of an average: add up all the things you have, and divide by the number of things. (The result in this example makes sense, because half the numbers are from 1 to 50, and half are from 51 to 100, so the average is half-way between 50 and 51.)
Similarly, a definite integral can also be thought of as a kind of aver- age. In general, if is a function of
, then the average, or mean, value of
on the
interval from
to
can be defined as
In the continuous case, dividing by b − a accomplishes the same
thing as dividing by the number of things in the discrete case.
Example
Show that the definition of the average makes sense in the case where the function is a constant.If is a constant, then we can take it outside of the integral, so
Example
Find the average value of the function for values of
ranging from 0 to 1.
The mean value theorem
If the continuous function has the average value
on the interval from
to
, then
attains its average value at least once in that interval, i.e., there exists
with
such that
.
The mean value theorem is proved in Proof of the mean
value theorem. The special case in which is known as
Rolle’s theorem.
Example
Verify the mean value theorem for on the interval from 0 to 1.
The mean value is 1/3, as shown in Example. This value is
achieved at , which lies between 0 and 1.
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