I can do multiple integration easily enough, but I am absolutely horrible at detecting symmetry. In fact, even if I know something is symmetrical I can't always discern what this means. In particular, I'm talking about double/triple integrals where some part of the equation simplifies to 0 due to symmetry.

For example, if we have

$\displaystyle \iiint (xy + z^2)dV$

Over the domain defined by

$\displaystyle 0 \le z \le 1 - |x| - |y| $

XY will cancel via symmetry. I understand if we have just X, or just Y and they are bounded by a domain that goes from negative to posative bounds that are equal, then yeah...they will be zero.

But how come XY = 0?

Similarly, if we have

$\displaystyle \iiint (3+2xy)dV$

over the hemisphere D given by

$\displaystyle x^2 + y^2 + z^2 \le 4$

$\displaystyle z\ge0$

This simplifies to

$\displaystyle \iiint (3)dV$

I simply do not see how. I know we are symmetric about the X and Y plane, sure, but what does this actually mean?

Cheers!