In[]:=
Assuming[{z>0,a>0},2*Pi*G*ρ*a^2*Integrate[Cos[θ]*(a*Sin[θ]-z)/(a^2+z^2-2*a*z*Sin[θ])^(3/2),{θ,-Pi/2,Pi/2}]]
Out[]=
2
2
a
Gπρ(-1+Sign[a-z])
2
z