The Area Problem

"Think of area as length times width"
It is then possible to find the area of any region with straight line borders.
The Method of Exhaustion was used by Archimedes to show that the area under a parabolic arch is 4/3 the area of the inscribed triangle by "exhausting" the difference between the two quantities.
The Fundamental Theorem of Calculus provides an easier, more generalizable way of computing areas in a manner similar to a heuristic that Archimedes used but did not trust. [Boyer pps. 50 - 52]