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Calc I - Final Paper [1]

# Calc I - Final Paper [1] - 1Chris Tomlinson Honors Calculus...

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1Chris Tomlinson Honors Calculus Professor Nguyen December 14, 2006 Integral Calculus vs. Differential Calculus Calculus is quite possibly the most important branch of mathematics. Along with math, it is used extensively in physics, engineering, astronomy, and many other sciences. Numbers and/or variables that are constantly fluctuating can be determined by this powerful branch of math, in particular cases that involve rates of change over time or area under a curve. During the end of the 17 th Century, the study of calculus really began to take off. Isaac Newton and Gottfried Leibniz were the two main mathematicians responsible and credited for the creation of infinitesimal calculus (even though Leibniz is credited with the name as Newton wanted to call it “the science of fluxions”). Today, there are two versions of calculus: integral calculus, which was created by Leibniz, and differential calculus, which was created by Newton. Integral calculus is the branch of calculus focusing mainly on areas under a curve. The Greeks were the mathematicians who took the first steps towards integral calculus using methods of exhaustion. The method of exhaustion is a way to find the area of a figure by inscribing a sequence of polygons, which in turn, converge to the area of the figure. Archimedes made probably the most significant Greek contribution with his method of exhaustion when he showed that the area of a segment of a parabola is 4/3 the area of the triangle with the same base and vertex and 2/3 the area of the circumscribed parallelogram. Using an infinite number of triangles progressively getting smaller and

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smaller filling in the voided space, Archimedes found the area of the segment of the parabola to be: A = area of triangles A (1 + 1/4 + 1/16 + 1/64 + … + 1/(4^n)) = (4/3)A. This method of exhaustion became the first known example of the summation of an infinite series. More important to integral calculus, however, was when Archimedes used this method of exhaustion to approximate the area of a circle. A diagram of his diagram is shown below: No further advancements on calculus were made until the 16 th Century. Mechanics and particular questions surrounding mechanics were posed in Europe during this time period, rejuvenating this study. Many mathematicians (Kepler, Fermat, Roberval, Cavalieri, and Huygens) made advancements during this time period on the subject as calculus was beginning to be made more rigorous. The advancements made
during this time proved to be the stepping stool for Leibniz and the creation of integral calculus.

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