RIEMANN-DARBOUX SUMS

Abstract

What is area? We are all familiar with determining the area of simple geometric figures such as rectangles and triangles. However, how do we determine the area of a region R whose boundary may consist of non rectilinear curves, such as a parabola?  To see how this could be done let us consider the following process. Suppose that a function  is continuous and non-negative on an interval  . We wish to know what it means to compute the area of the region R bounded above by the curve  below by the x-axis, and,  on the sides, by the lines  and  ,  in short, the area under the curve  as seen in the figure below.