$$int_0^2 int_0^sqrt{4-x^{2}} int_0^sqrt{4-x^2 -y^2} z sqrt{4-x^2 -y^2} , dz , dy , dx$$

The task is to solve this integral using spherical coordinate. After I tried to change the variable, I got

$$ int _0^{frac{pi }{2}}int _0^{frac{pi }{2}}int _0^2left(rho :cosleft(phi right)sqrt{4-rho ^2left(sinleft(phi right)right)^2}right):rho ^2sinleft(phi right)drho :dtheta :dphi

$$

Which I think pretty ugly with $sqrt{4-rho ^2left(sinleft(phi right)right)^2}$ . Is there anything I did wrong on the variable changing process? If it’s not, what are the approaches to solve this integral?

The integral simplifies like so

$$int_0^{2}int_0^{frac{pi}{2}}int_0^{frac{pi}{2}}rho^3sinphicosphisqrt{4-rho^2sin^2phi}:dtheta:dphi:drho = frac{pi}{4}int_0^{2}int_0^{frac{pi}{2}} rhosqrt{4-rho^2sin^2phi} :d(rho^2sin^2phi):drho$$

$$= frac{pi}{6}int_0^2 -rho left[4-rho^2sin^2phiright]^{frac{3}{2}}biggr|_0^{frac{pi}{2}}:drho = frac{pi}{6}int_0^2 8rho-rho(4-rho^2)^{frac{3}{2}}:drho = frac{8pi}{5}$$