Re: Info and a Question
Jim Francis (jfrancis@EDCC.CTC.EDU)
Mon, 19 Aug 1996 08:35:33 -0700
>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?
>
Brian:
How about the argument that you're trying to compare apples with oranges!
Many people have this intuitive notion that length and area and volume are
comparable quantities, but we know they're not. This apparent paradox
might be an excellent launching pad to briefly introduce Fractals
(Mandelbrot set, Koch snowflake, or maybe the Cantor set on [0,1]). The
other notion I believe is at work here has to do with our "limited"
understanding about the concept of infinity. I sometimes caution my
students about "playing with infinity" and how sometimes we can fool
ourselves about what's really going on.
Jim Francis ******* * * * * *
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