use cylindrical coordinates to evaluate the triple integral that gives the mass the solid

Question

use cylindrical coordinates to evaluate the triple integral that gives the mass the solid lying under the cone z=10-√(x²+y²), and above the xy-plane, if the density p is given by p(x,y,z)=x2+y2+z². Also, set up integrals to find the center of mass of the solid.

Answer 1

I'll start with the mass.  Substituting polar, we have z = 10 - r.  ρ = r² + z².  Also not that the cone intercepts 0 at radius 10.  So we have 0 < z < 10 - r, 0 < r < 10, and 0 < θ < 2π.  So we take the integral, with these bounds of:

 

(r² + z²) r dz dr dθ   (Note that the extra 'r' is from the conversion to polar.)

 

in order to find the mass.  

 

In order to find the center of gravity, we first have to find the moment.  We know that it is 0 along either x or y axis, so we simply have to consider the z axis.  So, over the same bounds, we take the integral of:

 

z(r2 + z2) r dz dr dθ

 

This provides us the moment, which we can divide by the mass (from the first step) in order to find the z-value of the center of gravity.