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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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View Full Document 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.
Leibniz spent much time in Paris acquiring the knowledge and textbooks of
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This note was uploaded on 10/24/2008 for the course MATH 1 taught by Professor 1 during the Spring '08 term at Rowan.
 Spring '08
 1
 Differential Calculus

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