Math Is Fun Forum

mikau

Member

Registered: 2005-08-22

Posts: 1,504

Gabriel's Horn, paint paradox

Today I heard something in class that at first amazed me, but later I began to question it.

The region under the curve y = 1/x and the first quadrant is, as we know, divergent. However, if we rotate this region about the x axis, we form a "horn" who's volume is pi  ∫  1/x^2 dx from 0 to infinity, which happens to be convergent.

My teacher further elaborated that we could fill the "horn" with a finite amout of pain, but if we dumped the pain out and looked inside, we would not find the inside to be fully painted. And the reason for this was that the surface area of the horn was infinite, but its volume was finite. This blew my mind for a while but later on I realized it was incorrrect. If we were to coat the sides of the horn with paint of a finite thickness it would be true, but we could in theory, paint forever if we used infinitly thin layers of paint, could we not?

So it doesn't seem like a paradox after or all. Or is it?

Anyway, just thought it might make an interesting discussion.

Last edited by mikau (2007-02-23 11:26:11)

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A logarithm is just a misspelled algorithm.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Gabriel's Horn, paint paradox

The region under the curve before you rotate it around the y-axis isn't the same as the surface area of the 3D space you get after the revolution, so the argument falls apart there as well.

But yes, I agree with you. Any amount of volume can always cover any amount of area, because it's got an extra dimension to play with.

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Why did the vector cross the road?
It wanted to be normal.