## Physical Intuition

In the previous lecture we mentioned a relation between velocity, distance, and the meaning of integration, which gave you a physical way of thinking about integration. In this section we generalize our previous observation.

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

Theorem 2.2.6 (Net Change Theorem)   The definite integral of the rate of change of some quantity is the net change in that quantity:

For example, if is the population of students at UCSD at time , then is the rate of change. Lately has been positive since is growing (rapidly!). The net change interpretation of integration is that

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:

Example 2.2.7   An ancient dragon is pacing on the cliffs in Del Mar, and has velocity . Find (1) the displacement of the dragon from time until time (i.e., how far the dragon is at time from where it was at time ), and (2) the total distance the dragon paced from time to .

For (1), we compute

For (2), we compute the integral of :

William Stein 2006-03-15