(* We compute the flux of F through a cone z=sqrt(x^2+y^2) oriented down *)

F={x,y,z^4}

(* parametrization *)
q={r Cos[t],r Sin[t],r}
qr=D[q,r]
qt=D[q,t]
n=Simplify[Cross[qt,qr]]

g=F/.{x->q[[1]],y->q[[2]],z->q[[3]]}
temp=Integrate[Dot[g,n],{r,0,1 }]
Integrate[temp,{t,0,2Pi}]