Bob leaves for a trip at 3pm (time *t* = 0) and drives with velocity miles per hour, where *t* is measured in hours. Find

and explain what it represents in the language of the problem.

Answer

First we find the value of the integral:

To explain what the integral represents in the language of the problem, we need to know units for *t* and units for the integral. We're told that *t* is measured in hours and *v*(*t*) in miles per hour. This means the integral is measured in miles. So

is the distance Bob drove from time *t* = 0 hours to time *t* = 2 hours. Since we're told 3pm corresponds to *t* = 0, *t* = 2 corresponds to 5pm. Putting everything together,

is the distance Bob drove from 3pm to 5pm.