[BUG] multiple integration

[Tevian Dray]

> One could argue that the pedagogical opportunity here goes back to the
> Fundamental Theorem of Calculus.  Why isn't \int_a^b f(x) dx equal to
> \int_b^a f(x) dx?  They both measure the area under the curve y=f(x)
> over the interval [a,b], don't they?

They differ by a sign, of course.  The (signed) area "under" the
curve y=f(x) would be obtained by integrating f(x) ds, or equivalently
f(x) |dx|, with either set of limits.

> Well, if you insist on the integral of a positive-valued function
> always being an area, then what does the classic FTC picture say about
> the derivative of F(x) := \int_a^x f(t) dt?
> 
> It says (and here's the chance for the students to review why the FTC
> works) that F'(x) = f(x) if x > a, but F'(x) = -f(x) if x < a.

NO IT DOESN'T!  The FTC most definitely says that F'(x)=f(x) for F(x) as
defined above regardless of whether x is bigger or smaller than a.  You
get a minus sign if you integrate from x to a, rather than from a to x;
the sign change comes from writing G(x) := \int_x^a f(t) dt, not from
whether x is bigger or smaller than a.

I suspect this is what you meant, but one shouldn't assume that the limits
on an integral are automatically reordered to put the smaller one first.

> Tevian's student certainly intended to evaluate \int g(y) dy on the
> interval (-\infty,0] (as one of the "slices" in an iterated integral);
> the student simply misapplied the Fundamental Theorem.  There's a nice
> lesson here about the fact that the integrals we compute using the FTC
> are oriented, and if you want dA = dx dy to make physical sense, you'd
> better be sure that your dx and dy integrals both have the orientation
> you want.

Yes, but I would argue that one should accomplish this by writing
	dA = |dx| |dy|
and choosing the signs on the right to match the given limits, rather than
by choosing the limits to ensure that dA = dx dy.

This is, I think the crux of the matter.  The assumption that the integral
of dA should be positive for both orientations contradicts the statement
that dA = dx dy.  Differential geometers might choose the latter, while I
suspect nonmathematicians would choose the former.  Either choice is fine.
But we seem to be teaching both, in my opinion misinterpreting Fubini's
theorem to argue away "inappropriate" limits.

Tevian