the graph of f(x)=1/x, rotated about the x-axis. finite volume (?), infinate surface area. but wouldnt the area of the outer surface = the inner surface? If you filled the horn with paint, wouldnt the inner surface area be covered?
I think that in this case, the volume is also infinite. But, I think that a horn with finite volume and infinite surface area could exist mathematically. Yes, and yes... but what's your point?
f(x)=1/x from [1,infinity) rotated about the x-axis does have finite volume and infinite surface area. If you have a bucket of 'mathematically perfect paint' then you can fill the horn with a finite volume of this paint. You can also paint it's exterior with this finite 3-d volume of paint with the understanding that my mathematical paint has no thickness when applied to a 2-d surface. Think what removing a 2-d cross section of paint from my bucket does to it's 3-d volume, nothing at all. If you are thinking of real paint, then it has thickness when applied and therefore a finite volume can't cover an infinite surface. However, this paint couldn't fill the horn either, it'd clog up the works when the horn becomes too thin.
shmoe is quite right mathematically speaking any finite amout of "paint" (ideal piant) can expane along a surface becomeing infintesimaly thin covering a sufficently large area, even an infinite area.
