FW: [BUG] multiple integration

[Smith, Alexander J.]

I guess one answer to when Delta A=dx dy is taught is in the details of
the hypotheses of Fubini's Theorem. Note textbooks tend not to attempt a
statement for the most general "measurable" region in the plane. Instead
the make the statement for a rectangle a<x<b, y<c<d or for a slightly
more general region such as a<x<b, g1(x)<y<g2(x). 

I have my students make extensive use of Matthias's Maple package
intdraw at this point in a course. If you try 
intdraw(x=0..1,y=1..0); then you get an error, which is good. It draws
attention to this issue that goes back the a hypothesis in Fubini. 

Alex Smith



-----Original Message-----
From: bug-bounces at science.oregonstate.edu
[mailto:bug-bounces at science.oregonstate.edu] On Behalf Of Tevian Dray
Sent: Monday, February 07, 2005 12:40 AM
To: Bridge Users Group
Subject: [BUG] multiple integration

I'm teaching multiple integration for the first time in several years.
I'm emphasizing the idea that integrals are sums, among other things by
the use of questions like:
	Is the integral of y^3 dA positive, negative, or zero
	over a region in the plane where y is negative?
Two unexpected (to me) issues have arisen, one of which I feel is simply
an error by the students, but the other is more subtle.

The first issue is the desire of many students to integrate before
answering the question, obtaining y^4/4, and arguing that this is
positive, so the integral must be positive.  This is simply a mistake,
but it does indicate how hard it is to get students to think about what
integrals mean without calculating anything.

The second issue is the subtle assumption that dA itself is positive.
This seems eminently reasonable when it really is an area, but is less
obvious in contexts with different units.  More importantly, issues of
orientation are being swept under the rug here.  One student explicitly
integrated over all negative values of y, but integrated from 0 to minus
infinity rather than the other way around.  Again, this is a mistake,
but where is this taught?  The only good explanation I know involves
surface integrals, where dA is the magnitude of the vector surface
element, and hence contains an absolute value.

Most texts hide this in the "definition" of multiple integrals as
iterated integrals, which has an orientation built in.  The instructor
can emphasize that the limits of such integrals always go from small
values to large ones, but that can cause problems later when dealing
with surface integrals, where there is no reason to make this
assumption.  In fact, I think it's a bad idea to argue, as some
mathematicians do, that one should always choose parameterizations of
curves and surfaces so that the parameters increase; better to teach
students to integrate from "start" to "finish" regardless of what that
implies for the parameters.  But how then should one introduce this
concept in multiple integration?

I think the answer must be to emphasize that the area element dA really
has an absolute value in it.  Whether this is done at the level of
Riemann sums, as
	delta A = |delta x| |delta y|
or in differential notation simply as
	dA = |dx| |dy|
or simply in words doesn't really matter, so long as dA is explicitly
assumed to be positive.

Comments welcome.

Tevian


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