Integration over a surface are

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When I start trying to solve this problem, I don't know where I go wrong. I think it should be in what I consider as my $r(\theta, \rho)$

$r(\theta,\rho)=(\rho cos(\theta), \rho sin(\theta), 0)$,
$0\leq \theta \leq 2\pi$ and $0\leq \rho \leq 1$

Which should be the correct r (I honestly don't get any of the answer choices)?

Thank you in advance for any help I can get.

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Not sure if correct, but I'll try:

I assumed z to be 0 everywhere, since the intersection's z-coordinate is zero everywhere, and then I used Green's theorem

Answer should follow easily to be d)