I want to calculate $\int_\Omega (curl f,n)dS$ for the area: $x=2rcos(t), y=2rsin(t), z=rcos(t)sin(t)$ with $0\leq r <1, 0\leq t <2\pi$ and vector field $(z,-z,y)^T$
I calculated $curl(f) = (2,1,0)$ and normal vector fiel $n=(\frac{-2rsin^3(t)}{\sqrt{4r^2+sin^2(t)cos^2(t)}},\frac{-2rcos^3(t)}{\sqrt{4r^2+sin^2(t)cos^2(t)}},\frac{4r}{\sqrt{4r^2+sin^2(t)cos^2(t)}})$ but as result of the surface integral I get $0$.
Can anyone tell me what I got wrong?