[BUG] multiple integration

[Greg Quenell]

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?

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.

But when we use FTC to evaluate integrals, we don't include this minus
sign, so we get oriented (or . . . what's the opposite of
coordinate-free?  Coordinate-bound?) integrals.  The moral is that the
FTC is a powerful tool, but it's not smart enough to give you a
physically meaningful answer if you put the limits in the wrong way
around.

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.


Sorry this is a little bit off the topic, but I wouldn't want to miss
an opportunity to mention the Fundamental Theorem in class, especially
to Calc III students, who are probably just about ready to understand
it.