The following is a restatement of the fundamental theorem of calculus:

change in number of students from time $t_1$ to $t_2$

Another very common example you'll seen in problems involves water flow into or out of something. If the volume of water in your bathtub is gallons at time (in seconds), then the rate at which your tub is draining is gallons per second. If you have the geekiest drain imaginable, it prints out the drainage rate . You can use that printout to determine how much water drained out from time to :

Some problems will try to confuse you with different notions of
change. A standard example is that if a car has *velocity*
, and you drive forward, then slam it in reverse and drive
backward to where you start (say 10 seconds total elapse), then
is positive some of the time and negative some of the time. The
integral
is not the total distance registered
on your odometer, since is partly positive and partly negative.
If you want to express how far you actually drove going back and
forth, compute
. The following example
emphasizes this distinction:

For (1), we compute

For (2), we compute the integral of :

William Stein 2006-03-15