ABSTRACT
We begin by studying what will be defined as the Riemann integral of a function f : [a, b] → R over a closed interval. Since the Riemann integral is the only type of integral we will discuss in detail,1 any reference to the integral of a function should be understood to be the Riemann integral of that function. One motivation for the development of the Riemann integral corresponds to the special case where such a function f is continuous and nonnegative on its domain. We may then determine the area under the curve described by the graph of f in the xy-plane lying above the x-axis. The key idea, so familiar to all students of calculus, is to approximate the area by a finite sum of rectangular subareas, a so-called Riemann sum, and then take the appropriate limit to obtain the exact area under the curve, the value of which corresponds to the value of the integral in this case. Even this special case prompts some natural questions. What other kinds of functions will obtain a limit in this process? Must the associated Riemann sum be set up in a particular way? Of course, we will be able to define such a Riemann integral even when the function f is negative over part of, or even all of, its domain. However, in such cases we usually forgo the interpretation of the integral as an area, instead relying on the more abstract notion of the integral as the limit of a Riemann sum.
