5 - Integral calculus

Abstract

The chapter first discusses the antiderivative, a function that possesses a given derivative. Integration as the limit of a summation process is defined and the process of constructing a finite increment in a function from knowledge of its derivative is described in the chapter. The chapter then discusses the role of the antiderivative as an indefinite integral and the use of tables of indefinite and definite integrals. Several methods of working out integrals without the use of a table are detailed in a section of the chapter. Finally, the use of integration to find mean values with a probability distribution is discussed. The antiderivative of a function is first described in the chapter. The process of integration is detailed. Indefinite integrals including tables of integrals are reviewed in another segment of the chapter. Also discussed in the chapter are improper integrals, methods of integration, numerical integration, probability distributions, and mean values. Integration is one of the two fundamental processes of calculus. It is essentially the reverse of differentiation, the other important process. There are several techniques for manipulating integrals into a recognizable form or that can be looked up in a table. Some integrals cannot be worked out mathematically, but must be approximated numerically. Some elementary techniques for carrying out this approximation are also discussed, including Simpson's rule, the most commonly used technique.