Math IA - Torricellis Trumpet During the 17th century student of Italian mathematician Galileo Galilei Evangelista Torricelli discovered a

Math IA - Torricellis Trumpet During the 17th century...

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Torricelli’s Trumpet During the 17th century, student of Italian mathematician Galileo Galilei, Evangelista Torricelli discovered a mathematical figure with an infinite surface area but a finite volume. This figure is seen as paradoxical as it does not have an end, as it stretches infinitely, but has a definite volume of π units. The figure discovered by Torricelli would be called Gabriel's Horn, but would also be known as Torricelli's Trumpet. The name given to the figure is a biblical reference to the horn that the archangel Gabriel will play to announce the commencement of Judgement Day, as the mathematical figure creates a similar shape to that of the horn. This figure is created by rotating the graph of the function y= 1/x , when x≥1 , around the x-axis. The function y=1/x has both a vertical asymptote at the y-axis which is done by setting x≥1 and a vertical asymptote at the x-axis. The asymptote at the x- axis is the reason why the figure stretches infinitely as it will get close to y=0 but will never touch. The primary reason I chose to explore Gabriel’s Horn or Torricelli’s horn was because of its name. I am a trumpet player for my high school, so when I read the name of the figure being Torricelli’s Trumpet I was automatically intrigued. Of course the name was not the only reason for the exploration but just what caught my attention when trying to choose a topic. Once I began doing research on Gabriel’s Horn I found the paradox associated with the figure interesting, which solidified my interest in writing my math exploration on this topic. I have always been interested in math that does not make sense at first glance. The final reason I chose to explore Gabriel's Horn was because 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. Revolutions Gabriel’s Horn is not a figure that can be found in nature, as it stretches to infinity. It is categorized as a solid of revolution, which are figures that start out with a function y=f(x) , on an interval [ a,b ]. The function is then rotated on either on the x-axis or y-axis to create a three dimensional region. In the case of Gabriel’s Horn, it starts out with the function y =1/x with the interval [ 1,∞ ] . The interval is important as this y=1/x is a rational function that provides a hyperbola with two curves: one in the first quadrant and another in the third quadrant. Once the function 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 integrals are used in order to find the volume and surface area of Gabriel’s horn. Before defining what improper integrals, I had to research what were integrals in general, as I did not have prior knowledge on the topic. The first integral that is taught in Calculus I is called an indefinite integral. Indefinite integrals are also known as the most general antiderivative of a function. So

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