The Generalized Method of Exhaustion

Abstract

AbstractThis first chapter introduces the Generalized Method of Exhaustion (GME) to represent a definite integral as an exact convergent series; it is not an approximation method such as the trapezoidal rule or Simpson’s rule; as such this method is indeed different from the ancient method of exhaustion. Integration, an important procedure in mathematics, leads to the representation of a function by a number, in general, by summing an infinite number of infinitesimals, in order to describe entities such as area of bounded region, volume of a solid, displacement and length of a curve. Some suggested references are Edwards (A treatise on the integral calculus with applications, examples and problems. Macmillan and Co., London, 1921); Ruffa (Int J Math Math Sci 31(6):345–351, 2002; Wave Motion 35(2):157–161, 2002); Strook (A concise introduction to the theory of integration, 2nd edn. Birkhauser, 1994). The chapter also presents an analytic derivation of the exact series representation of the definite integral. The method is appropriately illustrated by some interesting applications, such as a new derivation of the fundamental theorem of calculus.