Compute the flux integral where $F =<−0.5x^3 −xy^2 , −0.5y^3 , z^2>$ and S is the part of the paraboloid $z = 5−x^2 −y^2$ lying inside the cylinder $x^2 + y^2 \leq 4$, with orientation pointing downwards.
After all the simplifications, the flux integral can be simplified down to: $$\Phi = \int\int_{x^2+y^2\leq4}(x^2+y^2)^2dA$$ Converting this to the circular coordinate system: $$\Phi = \int_0^{2\pi}\int_0^2r^5drd\theta$$
However, simplifying this, the flux turns out to be $21.\bar3\pi$, which is apparently not the correct answer. Have I made a serious mistake in my calculations?