Finite existence $\int _A (y^2+(z-1)^2+x^2)^{\alpha} dx dy dz$

Question

$A=\{(x,y,z) \in R^3: x^2+y^2+z^2\le4;x^2+y^2+z^2\le 4z\}$

Can i use spherical coordinates$x=\rho \sin\theta \cos \phi,x=\rho \sin\theta \sin \phi,z=1+\rho \cos\theta$?