ABSTRACT
The derivative and the integral are the two central topics in a typical calculus course, but at first glance they have very little to do with each other. The integral is used to define the (net signed) area between the graph of a function and the horizontal axis. The Fundamental Theorem of Calculus allows us to compute a definite integral using antiderivatives. The Part 1 of the Fundamental Theorem is one of the most celebrated theorems in mathematics for its ability to tie together seemingly disparate ideas. The key step in proving the Part 1 of the Fundamental Theorem was to construct a Riemann sum in which the tagged points were determined by the Mean Value Theorem. Since derivatives are based on differences and integrals are based on sums, the essence of the fundamental theorem might be suggested by the fact that, loosely speaking, the sum of differences can be expressed as a difference of sums.
