# Determination Of The Decay Rate Of Radioactive Material Essay Example

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• Pages: 3 (678 words)
• Published: February 5, 2019
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Scientists use calculus to calculate the rate of decay of radioactive material, such as in nuclear weapons. Similarly, calculus helps determine how much fencing is required to surround an area and how gravity impacts objects in the field of Physics. Calculus is also utilized by mechanics to study the flow rates of fluids in vehicles. Integral calculus, a branch of mathematics that has evolved over time, was developed by ancient Greeks and Europeans. It continues to expand in modern times. The origin of mathematicians can be traced back to Egypt, where they discovered the volume formula for pyramids and approximations for the area of circles.

The Greeks later made significant breakthroughs in various areas. Archimedes, in particular, advanced the approach of using heuristic rules to expand upon the method of inscribed and circumscribed figures. These rules are problem-sp

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ecific and assist in directing the search process. Archimedes' arguments involved utilizing parallel slices of figures and the principles of lever laws, as well as viewing a surface as a composition of lines. Another notable discovery by Archimedes was determining the areas and volumes of figures through the application of conic sections (such as circles, points, hyperbolas) and evaluating infinitesimally thin slices of figures. This concept is still employed in integral calculus today.

Archimedes made a crucial discovery in integral calculus where he found a limit that allowed the "slices" of a figure to become infinitely thin. Euclid also contributed to the theory of calculus, but the logical basis was not supported at that time because infinity and continuity were not established. It was a mistake to believe that a definite integral could be found by adding up an infinite

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number of points, lines, or surfaces instead of taking the limit of an infinite sequence. However, these early discoveries were instrumental in aiding Newton and Leibniz in the development of calculus.

During the 17th century, individuals from various European countries made significant advancements in the field of integral calculus. Johannes Kepler demonstrated his understanding of integral calculus concepts in his Astronomia nova by using summations involving sine and cosine. His work was considered groundbreaking, with scholars acknowledging that he anticipated results found in integral calculus. Furthermore, F. B. Cavalieri built upon Kepler's contributions, particularly in the area of volume measurement.

During the 17th century in Europe, the study of calculus was greatly advanced. Pierre de Fermat made a significant discovery relating to finding the extreme values of algebraic expressions, which is now a fundamental concept in Differential Calculus. Additionally, Isaac Newton and Gottfried Wilhelm von Leibniz played important roles in the development of calculus. They defined calculus as the study of the relationship between a function and its derivative (Priestley 78).

Integral calculus, which is used to find areas and volumes under a curve, was initially developed by Newton. However, it was Leibniz who first published work on calculus in 1686. In contrast to Newton's symbols, Leibniz's notation involved using dy/dx to represent the derivative of y in terms of x. This distinctive notation serves as a reminder that the derivative can be understood as the limit of ratios of change or fractions.

Newton's work involved connecting infinite sums with algebraic expressions of the inverse relationship between tangents and areas. Before this discovery, rectangles were used to calculate area, but the estimated area was always either too small or

too large. Calculus allowed for these rectangles to become "infinitely thin" ("Integration"). Unlike algebra, calculus is able to find areas under a curve by working with continuously varying quantities of figures ("Calculus: Math in flux"). One of Leibniz's main interests was the properties of numerical sequences and the sum and differences of their terms. Blaise Pascal came very close to developing the fundamental theorem of calculus, which involves the derivative and the definite integral. Leibniz discovered this theorem partially due to Pascal's work in this area, and credit is also given to Cauchy.

The concept of the derivative and the definite integral being connected has been referred to as "the foundational idea of the entirety of calculus".

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