ABSTRACT
Problems involving the computation of areas and volumes of geometric figures date back to the some of the earliest writings [9, p.10], but the subject was first extensively developed by the ancient Greeks. It was the Greeks (Eudoxus 408 B.C.-355 B.C.) who developed the method of exhaustion, which computes the area within a geometric figure by tiling the figure with polygons whose areas are known. We begin by defining the area of a rectangle to be the product of its
length and width. Suppose then that F is a figure whose area is desired. The area of F can be estimated by comparing two constructions. First, cover the figure with a finite collection of rectangles so that the figure F is a subset of the union of the rectangles. The area of F will be no greater than the sum Ao of areas of the covering rectangles. Second, find a finite collection of rectangles which do not overlap, (except perhaps on the boundaries) and which lie inside F . The sum Ai of the areas of these interior rectangles is smaller than the area of F . For any such collections of rectangles,
Ai ≤ area(F ) ≤ Ao. This idea can be effectively used to compute the areas of a variety of shapes. Several specific area computations are discussed in this chapter. Af-
ter some simple cases illustrating Riemann sum calculations, the classical problem of computing the area π of a circle whose radius is 1 is considered. This problem was studied by the ancient Greeks. The next problem, the geometric development of the natural logarithm, was considered about two thousand years later. The final topic is Sterling’s formula, an approximation of n! which may be developed by geometric considerations and a bit of calculus.
