(* We compute the surface area of the upper hemisphere with radius R centered at the origin *) 
q={r Cos[th],r Sin[th],Sqrt[R^2-r^2]};
qr=D[q,r];
qth=D[q,th];
n=Simplify[Cross[qr,qth]]
nn=Simplify[Norm[n],Assumptions-> R≥0 && R≥r && r≥0 && th≥0 ]
Assuming[R>0,
Integrate[Integrate[nn ,{th,0,2 Pi}],{r,0,R}] 
]