Integral Calculus Essay

This essay has a total of 1462 words and 7 pages.

Integral Calculus

Ever wonder how scientists figure out how long it takes for the radiation from a nuclear
weapon to decay? This dilemma can be solved by calculus, which helps determine the rate
of decay of the radioactive material. Calculus can aid people in many everyday
situations, such as deciding how much fencing is needed to encompass a designated area.
Finding how gravity affects certain objects is how calculus aids people who study Physics.
Mechanics find calculus useful to determine rates of flow of fluids in a car. Numerous
developments in mathematics by Ancient Greeks to Europeans led to the discovery of
integral calculus, which is still expanding.

The first mathematicians came from Egypt, where they discovered the rule for the volume of
a pyramid and approximation of the area of a circle. Later, Greeks made tremendous
discoveries. Archimedes extended the method of inscribed and circumscribed figures by
means of heuristic, which are rules that are specific to a given problem and can therefore
help guide the search. These arguments involved parallel slices of figures and the laws
of the lever, the idea of a surface as made up of lines. Finding areas and volumes of
figures by using conic section (a circle, point, hyperbola, etc.) and weighing infinitely
thin slices of figures, an idea used in integral calculus today was also a discovery of
Archimedes. One of Archimedes's major crucial discoveries for integral calculus was a
limit that allows the "slices" of a figure to be infinitely thin. Another Greek, Euclid,
developed ideas supporting the theory of calculus, but the logic basis was not sustained
since infinity and continuity weren't established yet (Boyer 47). His one mistake in
finding a definite integral was that it is not found by the sums of an infinite number of
points, lines, or surfaces but by the limit of an infinite sequence (Boyer 47). These
early discoveries aided Newton and Leibniz in the development of calculus.

In the 17th century, people from all over Europe made numerous mathematics discoveries in
the integral calculus field. Johannes Kepler "anticipat(ed) results found… in the
integral calculus" (Boyer 109) with his summations. For instance, in his Astronomia nova,
he formed a summation similar to integral calculus dealing with sine and cosine. F. B.
Cavalieri expanded on Johannes Kepler's work on measuring volumes. Also, he
"investigate[d] areas under the curve" ("Calculus (mathematics)") with what he called
"indivisible magnitudes." A discovery by Pierre de Fermat on "finding the greatest and
least value of some algebraic expressions" ("Calculus (mathematics)") is now used in
Differential Calculus. Discoveries made in Europe at this time greatly helped the
development of calculus.

Later in the 17th century, Isaac Newton and Gottfried Wilhelm von Leibniz founded
calculus. Calculus is defined as the study of the interplay between a function and its
derivative (Priestley 78). Integral calculus is used to find areas and volumes under a
curve. Newton contrived calculus first, but Leibniz was the first to publish work on it
in 1686. Leibniz's symbols differed from Newton's; today, Leibniz used the notation dy/dx
to represent the derivative of y as a function of x, instead of Newton's notation y ˘.
This notation reminds people that the derivative is the limit of ratios of change and/ or
a limit of fractions. Newton's work included: linking infinite sums and the algebraic
expressions of the inverse relation between tangents and areas. Previous to their
discovery, rectangles were used to find area, though the estimated area was always too
little or too much; calculus allowed these rectangles to be "infinitely
thin"("Integration"). Algebra is not useful to find areas under a curve, unlike calculus,
which allows people to work with "continuously vary quantities" of figures ("Calculus:
Math in flux").

One of Leinbiz's main concerns was in the properties of numerical sequences and the sum
and differences of the terms in such sequences. Blaise Pascal came extremely close to
developing the fundamental theorem of calculus, which deals with the derivative and the
definite integral; his work in this area led Leibniz to discover this theorem partial
credit is given to him as well as Cauchy. The "relationship between the derivative and
the definite integral has been called 'the root idea of the whole of the differential and
integral calculus" (Boyer 11). Furthermore, he worked with sums and differences of
sequences to determine tangents, which is an important idea at the core of calculus
(Jesseph). Leibniz's Differential Calculus allows the problems of tangency to be reduced
to a relatively simple algorithmic procedure. This procedure allows for varying types of
curves to be studied. Another important detail of Leibniz's work includes using
antiderivatives; Leibniz found how to get a function's first derivative back to the
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