2. The integral, which turns out to be just the inverse of the derivative, historically dealt with the area under curves, the volume of fairly regular shapes, the work done in moving objects and the energy in light scattering, to name a few applications.
Both problems involve infinite sequences and their limits. The derivative deals with an infinite sequence of slopes of lines. The integral deals with an infinite sequence of areas under a curve (which in turn involves an infinite series).
Archimedes (217-21 B.C.) essentially invented the integral calculus. We'll start with this in Chapter 13 and find the area under curves different ways.