Find the mass of a bowl in the shape of a hemisphere
\[ x^2 + y^2 + z^2 = a^2, \qquad z\le 0 \]
if the density is $\delta(x,y,z)=1-z.$
Integrate $G(x,y,z)=xyz$ over the surface $C$ of the unit cube.
Find the flux of
\[ \F = yz\i - xz\j -xy \k \]
through the parabolic cylinder
\[ y=x^2,\quad 0\le x\le 1,\quad 0\le z\le 3, \]
oriented so that the normal vector points in the negative $y$ direction.
Find the flux of
\[ \F = x^2\i - yz\k \]
outward through the surface cut from the cylinder
\[ x^2 + z^2 = 4,\quad z\ge 0 \]
by the planes $y=0$ and $y=3.$
Find the center of mass of a conical band
\[ z^2 = x^2 + y^2,\qquad 1\le z\le 3 \]
with density $\delta = \frac1z.$