A Trombeta de Gabriel

Abstract

Infinity is a concept which often defies our intuition and leads us to making mistakes, for we have the idea that infinite is necessarily related to something unlimited. On Differential and Integral Calculus, for instance, we determine definite integral f x dxbaconsidering a continuous function f in a closed and limited interval [a, b]. However, in some applications we face cases where the interval is infinity or the function f has an infinite discontinuity in the interval. In both cases, we have an improper integral. Evidence of this problem have been observed in the seventeenth century, where in 1641, the Italian physicist and mathematician Torricelli noted that an infinite area, if submitted to a rotation around an axis of its plan, can sometimes provide a solid of revolution with a finite volume. Can something infinite generate something finite? This triggers a controversy about the nature of infinite and generates a real paradox. One of these fascinating solids of revolution is the Gabriel's Horn or Torricelli‟s Trumpet, generated out of an equilateral hyperbole, which can be enunciated as the Painter‟s Paradox and Gabriel's Horn: "If an infinite area bordered by the hyperbola xy = 1, the line x = 1 and the abscissa is rotated around the axis, the solid volume generated by this rotation is finite. Since this area is infinity, an infinite amount of paint would be necessary to paint it, however, a finite amount of ink would be enough to fill it, once the volume is finite." Intuitively, we could fill it with ink, but not even all the paint in the world would be enough to paint its surface. Without a doubt this is a counterintuitive example involving infinity. With that in mind, this paper aims to present through the Painter's Paradox and Gabriel's Trumpet an approach to teaching improper integrals for both, higher education and high school students who wish to deepen their studies on calculus. For this end, a content recall is done on subjects like length of a curve, surface of revolution area, solid of revolution volume and hyperbola. Furthermore, it is proposed a discussion about the importance of Calculus on basic education and the widely known "failure at teaching Calculus.