X[u_,v_]:={Cos[u],Sin[u],v}; F[{x_,y_,z_}]:={x^2,y,z}; 
Xu[u_,v_]:=D[X[s,v],s] /. s->u; Xv[u_,v_]:=D[X[u,t],t] /. t->v;
R[u_,v_]:=Cross[Xu[u,v],Xv[u,v]];
Integrate[F[X[u,v]].R[u,v],{u,0,2Pi},{v,0,1}]
