# Flux integral - parabloid inside a cylinder

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?

• Perhaps you could share your calculations. In any event I think this would be a good time to use Stokes' theorem instead of a direct calculation of the flux integral. – Umberto P. Apr 22 at 18:20
• @UmbertoP. Can you give me a hint as to how to use Stoke's theorem in such cases? Wouldn't we have to find an inverse curl? – Gummy bears Apr 22 at 18:55
• Ignore what I said. I read "curl" into flux integral without it actually being there. – Umberto P. Apr 22 at 19:01