Originally Posted by

**Deadstar** Using the divergence theorem, evaluate

$\displaystyle \int {\int}_\Sigma F.nds$,

where

$\displaystyle F = (4x, -2y^2, z^2), \Sigma$ is the bundary of the region defined by $\displaystyle x^2 + y^2 \le 4, 0\le z \le 3$, and n is the outward unit normal.

examples in notes only do this then F differentiates so that once the first integration is done, it can be simplified to something times $\displaystyle r = x^2 + y^2$, if you follow me. This one doesnt do that as far i can see so i dont know what to do...

My workings so far...

$\displaystyle \nabla F = 4 - 4y + 2z^2$

$\displaystyle \Rightarrow \int {\int}_\Sigma F.nds = \int \int {\int}_R (4 - 4y + 2z^2)dxdydz$

$\displaystyle = \int {\int}_D (12 - 12y + 18)dxdy$

$\displaystyle = \int {\int}_D (30 - 12y)dxdy$

Now is the point i was talking about where it has become simplified so that r may be used but i have no idea what to do here.

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