You're on the right track, but the mistake you're making is that the
paint can be infinitesimally thin in order to coat the surface. So, if the
thickness of the paint decreases proportionately to the function, then
you've only used a finite amount of paint (as the volume is only finite),
but you've coated an infinite surface area.
If someone happens to remember the exact way the integral are written,
that'd be a big help. I'm going to try and find my old Calc text now, I'm
sure it's in there somewhere.
gav
At 04:32 PM 2/9/00 -0800, Mike Bandsmer wrote:
>At 02:31 AM 2/9/00 -0500, gav wrote:
>> I think my favorite counterexample to arguments like this is Gabriel's
>>Horn. Take the function 1/x, and revolve it around the x-axis. You now
>>have something that looks very similar to a trumpet's bell. Now, find the
>>volume of this from 0 to infinity. It has a finite volume. However, it
>>has an infinite surface area.
>
>I have a little trouble conceptualizing what would happen if you fill this
>horn with paint. If you completely fill this horn with paint (a finite
>volume), the inner surface of the horn should be completely covered with
>paint, right? But the inner surface of the horn has infinite area, so
>wouldn't it take an infinite amount of paint to paint it? Where is my
>intuition going wrong?
>
>Mike
>
>
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