Find the center of mass of a uniformly solid cone of base 2a and height h.

Suppose the cylindrical symmetry of the problem to note that the center of mass must lie along the z axis (x = y = 0).

If the uniform density of the cone is ρ, then first compute the mass of the cone. If we slice the cone into circular disks of area pi r^2 and height dz, the mass is given by the integral:

M=∫ρdV=ρ∫pi r2dz from zero to h.

M=ρ∫pi a2 (1-z/h) 2dz from zero to h = pi a2 ρ∫ (1-2z/h+z2/h2) dz from zero to h = 1/3pi a2 h ρ from zero to h.

z=1/M∫ρzd V

z=1/4h