Numerical integral.

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Before Cavalieri introduced the system of indivisibles, Archimedes had also been working on the same thing. Archimedes had developed a method of calculating areas and volumes, but he was not willing to reveal his method. Elroy states that “Cavalieri developed a rational system of the so-called indivisibles and attempted to establish the validity of this approach” (63). Without the formalism of the integral itself, Cavalieri derived a number of basic theorems of integral calculus from his principles. Cavalieri’s method of calculation is accurate for parallelograms and plane figures lying between two parallel lines (Elroy 63). In 1639, Cavalieri obtained the power rule for integration of a polynomial, although the rule had been discovered three years before by Gilles de Roberval and Pierre de Fermat.

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A method for representing areas, lines, and volumes by indivisibles of a lower dimension than the continuous whole that is made by them, was thus developed by Cavalieri. Sriraman points out that “in his theory, Cavalieri stated that if two solids have the same height, then their volumes will be proportional to the areas of their bases” (7). The integration of f(X) =Xn was worked out through this method.

According to Wolfram Research, Inc, an integral is a mathematical item that can be defined as an area or an abstraction of an area (“Integral” Integration on the other hand is “the process of computing or obtaining an integral” (“Integration”). Integrals have been used in a number of ways namely:

1. In calculation of volume, area, mass and length: Integration is basically applied in Physics and Mathematics. With integration, it is possible to describe the curved area of a plane in terms of functions, and area and perimeter, in terms of integrals.

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