Two young mathematicians discuss what calculus is all about.

Check out this dialogue between two calculus students (based on a true story):
Ah. So now we have a connection between derivatives and integrals.
Right, the derivative of the accumulation function is the ‘‘inside’’ function.
So how do we use this to compute area?

Sometimes it helps to think about the most basic examples. Consider We know (by geometry) that this computes the area of a rectangle which equals . On the other hand, if we consider the accumulation function we see that

What is ?
On the other hand, the First Fundamental Theorem of Calculus says that if then . Armed with this knowledge, and the fact that , what must be?