>Consider the region bounded above by y = 1/x. Compute the area under
>this curve, above the x-axis, between x = 1 and x = 00 (infinity).
>The "area" of this region is given by fnInt(1/x,x,1,00) and is
>infinite (the improper integral diverges).
>
>However the volume obtained when we rotate this region about the x-axis
>is given by pi*fnInt(1/x*x,x,1,00) = pi.
>
>How do you expain to students that the area is infinite but the volume
>is finite, since the area fits inside the volume?
>
>Have fun! May your semester be full of the joy of teaching.
>
I think you've given away the farm when you set up the model
that the "area" is "inside" the volume. Bad idea. The area
is a well-defined measure of the surface that bounds the volume.
It's easy for me to imagine a finite volume, infinite area object
by thinking about a rubber ball, then a sponge of the same volume,
then a very rough sponge... I can even imagine a figure of
(essentially) zero volume that still has a (nearly) infinite
surface area. Imagine something like a hot water bottle with
lots more convolutions. Then take the limits to 0 and infinity.
William C. Mead
wcm@ansr.com
Visit "Adaptive Network Solutions Research, Inc." on the web
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