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Abstract
The purpose of this paper is to highlight and explain the contributions in Europe made towards modern integral calculus. Throughout this paper, I research contributions and ideas made by several mathematicians. These mathematicians include Archimedes, Kepler, Cavalieri, Fermat, Newton, Leibniz, Taylor, Euler, Simpson and Riemann. Beginning with the works of Archimedes on finding the areas of figures such as circles, spheres, and parabolas. This then moves onto Kepler, Cavalieri, and Fermat. Kepler and Cavalieri expand upon Archimedes’ work by finding the areas of a variety of figures. Fermat finds the maxima and minima of functions as well as finds the area beneath a specific function. Their ideas and theories are used by Newton and Leibniz, who define the Fundamental Theorem of Calculus on their own accord. Leibniz introduces modern notation to integral calculus. Taylor approximates nonpolynomial functions through polynomial functions. In addition, Taylor invents Integration by Parts. Euler summarizes infinite series, which allows for simpler integration. Simpson invents a variety of ways to integrate nonpolynomial functions. Riemann invents both the Riemann Sum and the Riemann Integral. The thesis concludes by explaining where the future of integral calculus is today.