Re: Horn of 1/X dilemma

Brablo wrote:

When you spin the function 1/X around on the X-axis,
and take the volume, you notice that the volume
is finite. However, the surface area is infiinite.

Another words, this shape can hold a finite amount
of paint, but it requires infinity gallons to paint it.

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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