# Z coordinates of a solid object moving

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 at the point (x,y,z) and occupies a region W, then the coordinates of the center of mass are given by Assume x, y, z are in cm. Let C be a solid cone with both height and radius 2 and contained between the surfaces and If C has constant mass density of 5 g/cm^3, find the z-coordinate of C's center of mass.

I can not find the right limits for and the integral function, so please help me to find it right

This is an upside-down cone, with the point at the origin, with surface $z = r$, and "base" of radius $r=2$ at height $z=2$.
I think of the $z$ integration as going from the surface $z=r$ up to the base at $z=2$. The radius goes from $0$ to $2$. The volume element can be taken as $2\pi r dr dz$. So we have $$\overline{z} = \frac{\int_{r=0}^2{\int_{z=r}^2 z\,2\pi r dr dz}}{\int_{r=0}^2{\int_{z=r}^2 2\pi r dr dz}} = \frac{4\pi}{8\pi/3}= {3\over 2}\,.$$
The $8\pi/3$ in the denominator is what you would get from the usual cone volume formula ${1\over 3}\pi R^2h$. The final answer is a point one-fourth of the way from the base to the point. You can check this on Wikipedia for example.