>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?
>
I haven't taught calculus in a million years, ... the fate of many a two
year college teacher, I suspect... so I haven't thought about this one for
ages. And I might have therefore remained silent, except that the responses
posted touched a nerve.
I guess that I think the posted responses don't really address the issue
for the average student. Or for me, I guess!
Thus, I can't help a (long) reaction to the posted responses, of course inviting
criticism and commentary in turn.
>A TYPICAL EXPLANATION I'D GIVE:
>1. (rhetorical question) Why is there any contradiction between infinite
>(2D) area and finite (3D) volume?
The contradiction is that you can imagine this horn shaped object which cannot
hold enough paint to paint its outside. Somehow this doesn't seem right!
>2. (followup) After all, the 1D "length" of the interval (0,00)
>("00"=infinity) is infinite while the 2D area under the exponential function
>on (0,00) is finite. (...and 'the length fits inside the area'?)
The same infinite length defines both the surface area and the volume.
>SEQUEL:
>The way I've heard the original dilemma posed is that finite paint will fill
>the "infinite can" BUT finite paint cannot cover its inside surface.
>A way out of this apparent dilemma comes from realizing that the paint
>canNOT be of (any) uniform thickness because the limit of the "can's"
>diameter=0.
I agree that this may approach a satisfying answer. By making the paint
thickness smaller and smaller, we can stretch that finite amount of paint to
cover as much of the surface of the horn as we wish.
>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.
If I picture a sponge of the volume of a rubber ball, itself made of rubber, and
then I melt down the sponge, I picture the volume of the puddle to be much less
than that of the ball. So I'm not sure which is the true volume of the sponge.
> 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 we referred to is of course mathematicians. Everybody else knows that length
and area and volume are indeed comparable quantities - in the "real world." And
I agree that this is a nice launching pad for fractals or the Cantor set, etc.
but these all appeal to the concept of limit, which is precisely what causes the
paradox in the original problem.
>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.
>
We do indeed have a limited concept of infinity, because it only exists as a
concept. There is no model of infinity in the universe, as far as I can
(literally and figuratively) see. The models are only in pure mathematics.
Hence the problem of explaining this to non mathematicians, or to fitting
mathematics with the concept that "God is a mathematician."
In fact, we have a terrible concept of area alone, never mind volume. What is
the surface area of a one inch square piece of polished steel, vice a one inch
square of that sponge? How minutely do we examine the surface, to measure its
area? In fact, if we get down to the atomic level, it seems to me that we can't
do better than some form of average surface area since all those molecules and
atoms are always moving anyway. And that has to somehow ignore the fact that
most of an atom is empty space - at least in the typical model we use.
My response to all of these quandaries is that it is dangerous to model the
physical universe using the concept of limit. (I recognize that mathematical
physicsts do this all the time - often ignoring inconvenient divergent limits,
singularities, etc.) We make certain definitions, especially of limits, that
allow us to go on and apply mathematics to "reality" without getting lost in the
details of the mathematics, such as 0.9999... = 1. "1" is much easier to work
with than that infinite geometric series, so we make that definition. And "1"
can be modeled in "reality," but not 0.9999... But I have never been sure that
0.9999... really IS unity. In fact, I don't think it is. In fact I have no idea
what 0.9999... is, except as a concept.
So, when we talk about infinite surface area we have already dug a hole,
whatever the context, by just mentioning the word infinite as a modifier of
something for which we think there is a physical meaning - surface area.
For this particular problem, and for the audience likely to receive it, I'll go
with an expansion of an idea mentioned above, that we can paint as much of the
horn as we want, if we make the paint thin enough. So, I guess that in the
limit, we can paint the silly thing!
By the way, a simpler example I like to cite of how infinite "things" behave in
absurd ways, is of the infinite hotel which is full, but can still offer a room
to someone asking for a room, by asking the person in room 1 to move to 2, and
that person to move to room 3, etc. No one is ever left roomless. My students at
any level like that one, which is taken from an old book entitled, I believe,
"Stories about Sets," a translation of some excellent Russian expository writing
about mathematics.
Philip Mahler
Middlesex CC
Springs Road
Bedford, Massachusetts 01730