The Sehnert lecture is next Monday: I
hope that you'll attend. You can still get in on the dinner. Let me
know if you would like to join us for dinner. Several of you have
signed up, and I'm glad!
Who's signed up?
Your quiz is returned.
Fubini! Fubini's theorem: If $f$ is continuous on the rectangle $R=\{(x,y)|a\le{x}\le{b},c\le{y}\le{d}\}$ then
\[
\int\!\!\!\int_{R}f(x,y)dA = \int_{a}^{b}\int_{c}^{d}f(x,y)dydx = \int_{c}^{d}\int_{a}^{b}f(x,y)dxdy
\]
This is true even if $f$ is merely bounded on $R$ and if $f$ is discontinuous only on a finite number of smooth curves.
My attempts at generalizing the trapezoidal rule did not
resonate, apparently: only a few of you noted the fundamental
idea, which is that we integrate a linear approximation to a
function (or data) to get a scheme (a "mask" -- which ends
up being the average of rectangular methods, just as it is in
the univariate case).
Hopefully you know the mask now that we might use for
rectangular data:
We're stepping up the dimension -- no big deal, but we've got more
work to do. Sometimes it's just three times as much work (as when we
have a separable function, a product of three functions
$f(x)g(y)h(z)$).