how to calculate this surface integral from vector integration

Question

i was having problem which was solved by stoke's theorem in that $\bar F =xy i-x^2j+(x+2)k$ and surface was $2x+2y+z=6$ in the first octant. so i calculated $$curl \cdot \bar F=-j+xk$$ and $$\bar N= \frac {2i+2j+k}{3}$$ $$\iint\bar N\cdot (\nabla \times \bar F)ds=\frac {1}{3}\iint(-2+x)ds $$

Answer 1

You can use Gauss theorem if the surface has a volume. or you can use Green theorem to convert the double integral into single-integral and then you have to find a vector Field that satisfies the theorem. these are the best 2 options you have..