Re: Gabriel's Horn
- From: "Dave L. Renfro" <renfr1dl@xxxxxxxxx>
- Date: 12 Oct 2006 07:05:19 -0700
Mkajuma Mbogho wrote:
Here we go again. I can prove that the surface area is infinite
and that the volume is finite. I can not however see how these
results can be possible. After all shouldn't the paint required
to fill the horn be at least as much as the paint needed to paint
the surface--since if you fill the horn with paint you DID paint
the (inner) surface. Infinitey is very very strange. I love it!!!
Now can someone PLEASE explain what is going on in this case.
For some reason, this never bothered me. The first time I heard
about it, I just considered it to be a higher dimensional version
of the similar fact that a curve in a bounded region can have
an infinite length, such as y = sin(1/x) or y = x*sin(1/x)
for 0 < x < 1 [but not y = (x^a)*sin(1/x) for a > 1, however].
Here's something I posted back on May 22 of this year:
Volumes are computed by multiplying three dimensions together,
areas by multiplying two dimensions. When you let all but one
of the dimensions get smaller and smaller, it's easier to get
convergence when you have TWO small quantities among your
factors (the volume computation) than when you have just
ONE small quantity among your factors (the area computation).
Interestingly, very few people seem surprised at the
same phenomena one dimension lower, where a curve of
finite length can bound a finite area. For example,
the area in the first quadrant bounded by x = 0,
x = 1, y = 0, and y = x + x*sin(1/x).
Dave L. Renfro
.
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