Trying to find the flux of a curl?

Question

I got this question asking for the flux of a curl F = (-2x^2)yi + 4xy^2j + z^2k, and is contained by the shell 2 - (x^2 + y^2) = z
above the xy plane.

I went through it but my answer came out weird. Any suggestions?

Answer 1

One suggestion is to use the divergence theorem to check your answer. It says the the flux of a vector through a closed surface is equal to the volume integral of the divergence of vector, and of course the divergence of a curl is zero. Hence you can find the flux through the shell above the plane by taking the opposite of the flux through the surface at z = 0, since they have to sum to zero.