Chapter 1 - Introduction

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This chapter provides an overview of numerical integration. Numerical integration is the study of how the numerical value of an integral can be found. The beginnings of this subject are to be sought in antiquity. A fine example of ancient numerical integration, but one that is entirely in the spirit of the present volume, is the Greek quadrature of the circle by means of inscribed and circumscribed regular polygons. Over the centuries, particularly since the sixteenth century, many methods of numerical integration have been devised. These include the use of the fundamental theorem of integral calculus, infinite series, functional relationships, differential equations, and integral transforms. Despite, or perhaps because of, the simple nature of the problem and the practical value of its methods, numerical integration has been of great interest to the pure mathematician. The most superficial glance at its history will reveal that many masters of mathematics have contributed to this field. Numerical integration is usually utilized when analytical techniques fail. When used sensibly and with proper controls, numerical integration can provide satisfactory answers. When used in a blind fashion—the availability of sophisticated computer programs makes it a temptation to operate blindly—numerical integration may lead to serious errors. Powerful symbolic integration systems have been incorporated into symbolic and algebraic computation systems as REDUCE, MACSYMA, and SCRATCHPAD.