7.1.8 Average Value of a Continuous Function

We began this section with a question about averaging a continuously varying temperature function. You saw that you could find a good estimate by adding a large number of closely spaced temperature values and dividing by the number of them. But how do you get the answer when there are literally infinitely many of them?

We actually answered that question in the distance-from-speed calculation, for which we already had a meaning for average speed. In fact, we know that

total distance traveled `=` (average speed) `times` (length of time interval),

or

average speed `=` (total distance traveled)/(length of time interval).

In our new language of integrals, the total distance is the definite integral of the speed function. Thus

average speed ${\displaystyle =\frac{1}{b-a}{\int }_a^{b}f\left(t\right)dt}$

where time runs from `t=a` to `t=b`.

Checkpoint 4

Definition   The average value of a continuous function `f` on an interval `[a,b]` is ${\displaystyle =\frac{1}{b-a}{\int }_a^{b}f\left(t\right)dt\mathrm{.}}$

Checkpoint 5