The paraboloid equation can be written x^2+y^2=4-z. At z=0 this is the circle x^2+y^2=4 (centre at the origin and radius=2). As we move along the z axis the cross-section parallel to the x-y plane remains as a circle (x^2+y^2≤4) which reduces to a point at the vertex of the paraboloid at (0,0,4). The centre of gravity (COG) is therefore on the z axis at P(0,0,p) because the paraboloid is symmetrical, with the z axis acting like a spindle. We have to find p.
We can consider the solid as a set of discs pierced through their centres by the z axis as a spindle. The volume of each disc is related to its mass by the constant density. The thickness of the disc is the infinitesimal dz. The radius of a disc is √(4-z), so its volume is π(4-z)dz. This expression can be taken to represent its mass.
The mass of each disc is concentrated at its centre because of symmetry, so its COG is at the centre on the z axis. We need to consider the COG of all the discs together, so we need to define the moment of each disc about P. This is the product of mass and distance from P, which is p-z. The negative moments have to balance the positive moments. The sum total of the moments is zero. The expression for the moment of each disc is π(4-z)(p-z)dz and the sum of these must be zero, so π∫((4-z)(p-z)dz)=0 where the limits are 0≤z≤4.
π∫((4p-z(p+4)+z^2)dz)=0; π[4pz-z^2(p+4)/2+z^3/3]=0, 0≤z≤4.
π(16p-8(p+4)+64/3)=0; 8p-32+64/3=0; 8p=32/3, p=4/3, so the COG is P(0,0,4/3).