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 $$\overline{x} = \frac{1}{m}\int_W x\rho \, dV \quad \overline{y} = \frac{1}{m}\int_W y\rho \, dV \quad \overline{z} = \frac{1}{m}\int_W z\rho \, dV,$$

Assume $x$, $y$, $z$ are in cm. Let $C$ be a solid cone with both height and radius 2 and contained between the surfaces $z=\sqrt{x^2+y^2}$ and $z=2$. If $C$ has constant mass density of 3 g/cm$^3$, find the $z$-coordinate of $C$'s center of mass.

$\overline z =$
(Include units.)