Gabriel's Horn

Date: 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/