Numerical integral - Research Paper Example

Mathematics 13 November Numerical Integral The history of integrals s back to the 17th century. The basis of integrals was formed in about 1635 by an Italian mathematician known as Cavalieri. His work was the fundamental modern step towards the branch of calculus. However, the development of integrals has had many contributions by different mathematicians. Before the development of integral calculus by Gottfried Wilheim Von Leibnz and Sir Isaac Newton, several other mathematicians had made several developments using the ideas of Archimedes of Syracuse (Elroy 63).

Cavalieri was one of the pioneer proponents of the idea that quantities too small cannot be divided into halves, in a concept known as the concept of indivisibles. 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.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” Wolfram.com). 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. Integration is also used to calculate the mass of an object where the density of the matter within it can change from place to place (“Integrals at Work”).2. In launching Spacecrafts: Integration is applied in solving equations that yield the most excellent aerodynamic shape for the body of a spacecraft.3. In calculation of population growth: Systems of equations used to calculate population growth apply integration at some point in their solutions (“Integrals at Work”).

Integration has made it possible to solve significant problems in physics, mathematics and astronomy. Today, it is applied in the fields of medicine, linguistics, music, and computer technology.Works Cited“Integrals at Work.” The Wolfram Integrator. n.d. Web. 13 November 2012. McElroy, Tucker. A-Z of Mathematicians. New York: Facts on File, Inc., 2005. Print.Sriraman, Bharath. The Montana Mathematics Enthusiast Monographs in Mathematics Education. Missoula: Information Age Publishing Inc. & The Montana Council of Teachers of Mathematics, 2012. Print.Wolfram Research, Inc.

“Integral” Wolfram.com. 2012. Web. 13 November 2012.Wolfram Research, Inc. “Integration” Wolfram.com. 2012. Web. 13 November 2012.