Integral Calculus Essay, Research Paper
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 original function. For example, knowing an object’s velocity and the time it travels, you can find its acceleration.
Though the discovery of calculus was outstanding, some problems existed in Leibniz’s calculus. One such problem is how to detect the motion of a vibrating string. Leibniz’s students, the Bernoulli family of Swiss mathematicians, used calculus to solve this problem. Also they used calculus to solve other problems, such as finding the curve of quickest descent connecting two given points in a vertical plane.
After the development of calculus, new ideas were and are still being developed. In the 18th century, Leonhard Euler, a Swiss-Russian, wrote on new ideas such as an analytic approach of trigonometric and exponential functions, furthering the development of calculus. Another significant discovery in integral calculus was that A. L. Cauchy gave a “secure foundation” (”Calculus (mathematics)”) to calculus by his theory of limits. Calculus now had a “secure foundation” because Cauchy, in addition to his theory of limits, he stated a function must be continuous (Grattan-Guiness 361) and convergent (”Cauchy” 434). Some of Cauchy’s other discoveries are defining upper and lower limits, “proving a continuous function has a zero between [endpoints] where its signs are different,” either positive or negative (”Cauchy” 434).
More work was done in this field by N. Lobachevsky, P. G. L. Dirichlet, and J. W. R. Dedekind on functions. These mathematicians found a definition of a function. Another mathematician, G. F. B. Riemann, worked with Archimedes’ idea of sandwiching the area above and below as partitions become finer (Grattan-Guiness 361). Jordan worked with Darboux’s use of upper and lower sums to avoid mistakes found with Riemann’s work. One of these mistakes included that the same value was given to the content of a set P as to its closure (same limit points). A solution was needed to satisfy both the measure of a set and the integral of a function, which was a domain. Another problem discovered was giving a definition of the measure of a set, and the integral of a function, which satisfied them all (Grattan-Guiness 363). Lebesgue has strived to solve this problem using a more general approach than mathematicians had made in the past (Grattan-Guiness 364).
Recently, Tom Hern, Cliff Long, and Andy Long took a look at integral calculus. They dealt with double integrals, dealing with two integrals at the same time, and order of integration. For example, if there were two integrals containing x and y, the order of integration determines which variable is integrated first. Usually the order of integration is ignored, but when dealing with certain double integrals, it is necessary to follow the order, such as with the Hoffman surface, a figure constructed in using double integrals (Hern et al.).
Without the Egyptians’ discoveries with volumes, Archimedes’s work with limits, or Europeans’ numerous advances in measuring volumes, Newton and Leibniz never would have discovered calculus. Many advancements have been made since, including Cauchy’s advances in function theory and double integrals. When first developed, neither man knew how much calculus would simplify everyday problems in today’s world, such as acceleration and the effect of gravity on an object.
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