(* We compute the surface area of the upper hemisphere with radius R centered at the origin *) 
r={R Sin[ph]Cos[th],R Sin[ph]Sin[th],R Cos[ph]};
rt=D[r,th];
rp=D[r,ph];
n=Simplify[Cross[rt,rp]]
nn=Simplify[Norm[n],Assumptions-> R>0 &&0<th<2 Pi&&0<ph<Pi/2]
Integrate[Integrate[nn,{th,0,2 Pi}],{ph,0,Pi/2}]