The motion of a solid object can be analyzed by thinking of the mass as concentrated at a single point, the center of mass. If the object has density $\rho(x,y,z)$ at the point $(x,y,z)$ and occupies a region $W$, then the coordinates $(\overline{x},\overline{y},\overline{z})$ of the center of mass are given by $$\overline{x} = \frac{1}{m}\int_W x\rho \, dV \quad \overline{y} = \frac{1}{m}\int_W y\rho \, dV \quad \overline{z} = \frac{1}{m}\int_W z\rho \, dV,$$ where $m=\int_W \rho\,dV$ is the total mass of the body.

Consider a solid is bounded below by the square $z=0$, $0 \le x \le 2$, $0 \le y \le 3$ and above by the surface $z= x+y+1$. Let the density of the solid be 1 g/cm${}^3$, with $x,y,z$ measured in cm. Find each of the following:

The mass of the solid =
$\overline x$ for the solid =
$\overline y$ for the solid =
$\overline z$ for the solid =
(For each, include units.)