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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?

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  • $\begingroup$ 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. $\endgroup$ – Umberto P. Apr 22 at 18:20
  • $\begingroup$ @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? $\endgroup$ – Gummy bears Apr 22 at 18:55
  • $\begingroup$ Ignore what I said. I read "curl" into flux integral without it actually being there. $\endgroup$ – Umberto P. Apr 22 at 19:01

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