Symmetry of the domain? For the tiple integral 
$$\iiint_D (z^2+z) \,dx\,dy\,dz$$
over the domain $D:x^2+y^2+z^2\leq4,\quad z^2\leq x^2+y^2$
The textbook states that by symmetry of the domain the integral simplifies to
$$\iiint_D z^2 \,dx\,dy\,dz.$$
How exactly do they arrive at that conclusion? 

This is because 
$$
    \iiint_D z\,dV = 0
$$
Why is that? If $(a,b,c)$ is in $D$, then so is $(a,b,-c)$ (both inequalities have $z^2$ in them).  So each point $P$ in $D$ has a mirror-image point $P'$ on the opposite side of the $xy$-plane.  If $f(x,y,z) = z$, then $f(P') = -f(P)$.  Accumulating all the points $P$ above the $xy$-plane, their $f$-values cancel in pairs with their corresponding points $P'$.  
