Paradox of the trumpet shape

This is a question I had for long time now, when you rotate the function $y=1/x$, $x>0$ (say $x$ and $y$ both measure meters) about the $x$ axes by $2\pi$ you get a shape which has infinite surface area and finite volume.Lets call this shape "trumpet shape". Now the weird thing is that suppose I have a "trumpet" shape that is made of by arbitrarily small transparent material (imagine folding a transparent sheet to this trumpet shape). Since it's volume is finite I can fill the whole trumpet with finite amount of paint say $k$ litres.But since the surface area is infinite, no matter how many paint I have I can still not paint it's surface area. Now suppose I pour $k$ litres of paint in my "trumpet shape", then the whole trumpet is filled with paint.Now imagine how this trumpet made of transparent material looks like.It looks like it's surface area is painted and since our transparent material is arbitrarily small we can effectively say that the surface area is painted using $k$ litres of paint. A contradiction
What am I saying wrong here?
Thank you

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This is sometimes known as the Gabriel's horn paradox. – David Jun 12 '14 at 1:21
Wouldn't the paint fall out the bottom of any finite approximation to this shape? – DWin Jun 12 '14 at 1:49

2 Answers

Your issue is trying to compare a 2 dimensional object (surface area) with a 3 dimensional object (volume). Any volume of liquid can be spread thin enough to cover as much surface area as you want (mathematically speaking, there are probably physical limits to this).

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cheers, I get it now – TheGeometer Jun 12 '14 at 1:49

Suppose you want to paint the surface with a coat of paint with a constant thickness. The volume of paint you will need is (surface area)*(thickness), which is infinite.

However, when you fill the inside with paint, the thickness of the coating is not constant, in fact it decays to 0 as x goes to infinity. This is why a finite volume of paint can fill the inside of the trumpet.

See the Wikipedia article on Gabriel's Horn for more information.

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