Two young mathematicians discuss whether integrals are defined properly.

Check out this dialogue between two calculus students (based on a true story):
Riley, I like integrals.
I feel fancy when I make an integral sign.
I know! An integral computes the signed area between a curve and the -axis. But why signed area? Maybe we should just compute plain old area.
Makes sense to me!
Unless…maybe there are other applications where ‘‘signed’’ area makes more sense.

One really great way to think about integrals is that they ‘‘accumulate rates.’’

Write down as many examples of ‘‘rates’’ and ‘‘accumulated rates’’ as you can. For example:

miles per hour is a rate, and miles is then an accumulated rate.