ABSTRACT

The Riemann Integral In this chapter we introduce the Riemann integral and deduce its salient properties. In all probability the reader is already familiar with the integral and its applications from earlier calculus courses. Despite the apparent technicality of the various definitions of the integral that we offer in Sections 8.1 and 8.2, the concept arose from our need to handle a simple and intuitive idea: how do we define and calculate the areas of shapes in the plane? Starting with a square whose side length is 1 as the unit area, in-

tuition suggests that the area of a shape should be the number of non-overlapping unit squares needed to cover that shape exactly. With this in mind, we quickly obtain laws for the areas of simple shapes like rectangles, triangles, and more general polygonal shapes. Significant difficulties arise, however, when we want to deal with shapes bounded by curves rather than straight edges. The first attempt in recorded history at working out such areas is credited to Archimedes (287-212 BC), if we overlook some evidence that the ancient Egyptians had quite accurately calculated the area of the circle. Archimedes determined the area under the parabola y = x2 over the interval [0, 1] by approximating it using polygonal shapes. This is the very idea on which the Riemann Integral, named after B.Riemann, is founded. Section 8.4 deals with the relationship between integration and dif-

ferentiation. The all important “Fundamental Theorem of Calculus”, derived independently by Newton and Leibnitz, reveals in striking simplicity how one operation is essentially the inverse of the other. This is rather surprising, since the derivative of a function measures its rate of change, or the slope of its curve, while its integral measures the area under its curve, and there is no apparent connection between the two concepts.