Gabriel's HornDate: 2/15/96 at 16:44:0 From: Anonymous Subject: Gabriel's Horn How is it that the surface area of the horn is infinite while the volume is finite? Thanks for your time, Mi Date: 8/16/96 at 14:32:30 From: Doctor James Subject: Re: Gabriel's Horn There are two parts to answering this question. The first is showing it mathematically, and the second is explaining why it is not a problem. Gabriel's horn, the figure formed by rotating the graph f(x)=1/x, x>=1 around the x-axis, can have its interior integrated via the 'disks' method. Each disk will have a radius of y=1/x, and so will have an area of pi*x^(-2). It will have a thickness of dx, so the integral, evaluated from 1 and the limit of positive infinity (+oo), is / -2 | lim +oo pi | X dX = -pi/X | = -pi*0 + pi/1 = pi, / | 1 which is finite. The surface area can be integrated by rings, with circumference 2pi*y=2pi/x, and width ds = Sqrt{1 + 1/x^4} dx (calculated using the Pythagorean Theorem). Again, integrating from 1 to lim +oo, we have / -1 surface area = 2pi | x Sqrt{1 + 1/x^4} dx / Note that Sqrt{1 + 1/x^4} is always at least as big as 1, so we can drop it from the integral, and if the resulting integral diverges to infinity, then the surface area integral diverges too. So let's do this integral: / -1 | lim +oo 2pi | x dx = 2pi*ln(x) | = 2pi* lim(ln(x)) - 2pi*0, / | 1 x->+oo which is undefined and approaches positive infinity, because ln(x) approaches infinity as x does. Why isn't this a problem? Well it certainly is weird to have an object with infinite surface area but a finite volume, but you have to be careful when you talk about infinities. The surface has infinite two- dimensional area, which does not have much to do with three- dimensional volume! Another example of something like this is if I have an infinite plane in three-dimensional space, covered with an infinitely thin rubber sheet, and you tell me to make a volume of less than or equal to 10 cubic units between the sheet and the plane. I can make a nice little bulge towards the center which has a volume of 5, which leaves me with 5 or less than 5 for the rest of the sheet. In successive rings from the center bulge, each ring being 1 unit wide, I can make the volume between the sheet and the plane be only half my remaining volume, since I can push the sheet as close as I want to the plane. And so on until the whole sheet is in position. But then you have a finite volume, covered by two things (the sheet and the plane) with infinite areas! -Doctor James, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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