Determination Of The Decay Rate Of Radioactive Material Essay Example

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".