Section Summary

In this section we have seen that we can estimate the average of a continuous function, such as temperature or speed, by adding a large number of closely spaced values and dividing by the number of them. The limiting value of these estimates as the number of terms in the sum becomes arbitrarily large is the area under the graph of the function divided by the length of the interval.

We approached the problem of area under the graph of a nonnegative function in a different way, by adding up areas of thin rectangular strips. In this case, the limiting value — summing infinitely many infinitely small areas — is the concept we named the definite integral of the function. Thus the average value of a continuously varying function can be described as the definite integral of the function divided by the length of the interval.