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Torricelli’s TrumpetDuring the 17th century, student of Italian mathematician Galileo Galilei, EvangelistaTorricelli discovered a mathematical figure with an infinite surface area but a finite volume. Thisfigure is seen as paradoxical as it does not have an end, as it stretches infinitely, but has a definitevolume of π units. The figure discovered by Torricelli would be called Gabriel's Horn, but wouldalso be known as Torricelli's Trumpet. The name given to the figure is a biblical reference to thehorn that the archangel Gabriel will play to announce the commencement of Judgement Day, asthe mathematical figure creates a similar shape to that of thehorn. This figure is created by rotating the graph of the functiony= 1/x, when x≥1, around the x-axis. The function y=1/xhasboth a vertical asymptote at the y-axis which is done by settingx≥1and a vertical asymptote at the x-axis. The asymptote atthe x- axis is the reason why the figure stretches infinitely as itwill get close to y=0but will never touch. The primary reason I chose to explore Gabriel’s Horn orTorricelli’s horn was because of its name. I am a trumpet playerfor my high school, so when I read the name of the figure being Torricelli’s Trumpet I wasautomatically intrigued. Of course the name was not the only reason for the exploration but justwhat caught my attention when trying to choose a topic. Once I began doing research onGabriel’s Horn I found the paradox associated with the figure interesting, which solidified myinterest in writing my math exploration on this topic. I have always been interested in math thatdoes not make sense at first glance. The final reason I chose to explore Gabriel's Horn wasbecause the of the math used to find its volume and surface area. It uses Calculus that at the time
had not been covered, so I saw an opportunity to learn something completely new to me. RevolutionsGabriel’s Horn is not a figure that can be found in nature, as it stretches to infinity. It iscategorized as a solid of revolution, which are figures that start out with a functiony=f(x), on aninterval [a,b]. The function is then rotated on either on the x-axis or y-axis to create a threedimensional region. In the case of Gabriel’s Horn, it starts out with the function y=1/xwith theinterval [1,∞]. The interval is important as thisy=1/xis a rational function that provides ahyperbola with two curves: one in the first quadrant and another in the third quadrant. Once thefunction is graphed it is rotated around the x-axis until it provides a three dimensional figure. Indefinite Integrals Gabriel’s Horn is used as an application for what are call improper integrals. Improper integralsare used in order to find the volume and surface area of Gabriel’s horn. Before defining whatimproper integrals, I had to research what were integrals in general, as I did not have priorknowledge on the topic. The first integral that is taught in Calculus I is called an indefiniteintegral. Indefinite integrals are also known as the most general antiderivative of a function. So