High Number Punch 03

$A closed area occupied by a uniform object (density \ rho is constant) \ Omega is surrounded by a surface z = x ^ 2 + y ^ 2 and a plane z = 0, | x | = a, | y | = a.$
$(1) Find the volume of the object;$
$(2) Find the center of mass of the object;$
$(3) Find the rotational inertia of the object about the z-axis.$

$Solution: (1) As shown in the figure, from the symmetry of \ Omega$