Subject: Re: gabriel's horn
From: Martha Haehl (haehl@kcmetro.cc.mo.us)
Date: Fri Feb 25 2000 - 20:44:22 EST
This is not an intuitive explanation of Gabriel's Horn, but possibly adds to
the paradox. That an infinite area rotated can result in a finite volume.
The paradox comes from a Calc III look at Gabriel's horn.
If the volume is taken by calculating the limit of Riemann sums set up in
the following manner, the limit is one where at least one of the 3-d slices
(if not all) has infinite volume. The final limit of the Riemann sum would
involve a limit as the thickness (assuming same thickness for each slice)
goes to zero while the area of the slice is infinite.
Set it up this way. The z be the horizontal axis, the x-axis from left to
right, and the y-axis from front to back. z = | 1/x |, for x positive,
gives upper and lower bounds of the intersection of the horn with the x-z
plane. Partition the horn by taking cross-sectional areas with uniform
thickness "delta y" and the area between the upper and lower portions of the
curve, z = | 1/x |, created by cutting the horn with planes parallel to the
x-z plane. Then create the Riemann sums of the cross-sectional areas (which
is not really a Riemann sum since each term of the sum is infinite). Take
the limit of the sum as delta y goes to zero, and we know the result is
finite. I find this an interesting example of the indeterminant form,
(zero)(infinity).
Happy thinking!
Martha
----- Original Message -----
From: Paul Hertzel <hertzpau@niacc.cc.ia.us>
To: <mathedcc@archives.math.utk.edu>
Cc: Paul Hertzel <hertzpau@niacc.cc.ia.us>
Sent: Friday, February 25, 2000 12:33 PM
Subject: Re: gabriel's horn
> RayM wrote,
>
> > Once the
> >coating thickness is specified, it is trivial to calculate the area that
a
> >given volume will cover _to that thickness._ The comparison to 1/x is
> >that filling volume allows the "coating" thickness to thin out away from
> >the origin. Actually painting the surface requires the volume between
the
> >surfaces generated by 1/x and 1/x+(coating_thickness), i.e. an infinite
> >slab.
>
> If I'm understanding this correctly, then you're concluding an infinite
volume
> of paint is needed to paint the outside surface. But this is false. You
vary
> the thickness as you paint to the right, and there's more than enough to
go
> around. Make the thickness a function of the radius at a given point, and
you
> can paint the surface with an eyedropper of paint.
>
> You just can't paint it with a 2-d liquid, whatever that is. It's the
> ability to vary
> the 3rd dimension that is missing from the 2-d reality.
>
> Paul Hertzel
> NIACC
> Mason City
>
>
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