(* We compute a surface integral of the scalar field z+x+2y on the portion of the plane 2x+2y+z=2 in the first octant *) 
f=z+x+2y;
q={u,v,2-2u-2v};
qu=D[q,u]
qv=D[q,v]
n=Simplify[Cross[qu,qv]]
nn=Simplify[Norm[n]]
for=f/.{x->q[[1]],y->q[[2]],z->q[[3]]}
Integrate[for nn ,{v,0,1-u}]
Integrate[%,{u,0,1}]

ContourPlot3D[2x+2y+z==2,{x,0,1},{y,0,1},{z,0,2}]