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