[BUG] multiple integration

[Tevian Dray]

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