Hi folks
> >> 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.
> 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.
If y=f(x), the volume of revolution is given by
{integral from 1 to infinity) pi.y^2 dx
Note the integral starts at 1, not zero (otherwise the volume is undefined)
for the Horn. The volume is in fact pi.
The surface of revolution is given by
(integral from 1 to infinity) 2.pi.y.sqrt(1+y'^2) dx
where y'=dy/dx= -1/x^2 in the case of the horn.
The integrand is 2.pi.x /sqrt(x^4+1). If you recognise this, good for you
(Apply a change of variable t=x^2 and you will get pi. 1/sqrt(t^2+1) under
the integral, which you might recognize. If you're still stuck, think about
arcsinh t).
Recognize it or not, it really doesn't matter, Note the integrand is
actually a little greater than
2 pi y dx
which is the usual mistake first made with surfaces of revolution
(approximating the surface by 'delta-x height cylinders' instead of 'delta-x
height slices of cones'). However this function is a lot easier to recognize
as the derivative of 2.pi.ln x, and so the integral as we approach infinity
is indeed unbounded.
> >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?
It's a bit like the old gag "how many lawyers does it take to wallpaper a
room?" ("Depends how thinly you slice them"). A given amount of paint or
lawyers can cover an arbitrarily large surface provided you spread it thinly
enough. Not possible in the real world of course (not to mention the Horn's
neck is ultimately too narrow to squeeze a paint molecule down), but
mathematicians aren't limited by such physical constraints.
Single-celled organisms have known for eons that the best way to improve
their rate of nutrition is to stretch their volume into the largest possible
surface area. Fortunately physics intervenes and an infinitesimally thin
organism of infinite length but finite volume isn't a biological
possibility.
Chris Nash
Lexington KY
UNITED STATES
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