Mass, moment, and center of mass of a plate bounded by the x axis and y = 1 - x^2.

mass = Integrate[ x^2 * y, {x,-1,1},{y,0,1-x^2} ]

mX = Integrate[ y * x^2 * y, {x,-1,1}, {y,0,1-x^2} ]

mY = Integrate[ x * x^2 * y, {x,-1,1}, {y,0,1-x^2}]

{ mY/mass, mX/mass }

Moments of an upside-down pyramid.

rho[ x_, y_, z_ ] := E^z - x*y^2

mXY = Integrate[ z *rho[x,y,z], {z,0,1}, {x,-z,z}, {y,-z,z}]

mXZ = Integrate[ y *rho[x,y,z], {z,0,1}, {x,-z,z}, {y,-z,z}]

mYZ = Integrate[ x *rho[x,y,z], {z,0,1}, {x,-z,z}, {y,-z,z}]