So I'm supposed to calculate
$$ \iint_S (x^2+y^2)dS $$
Where $S$ Is the part of the plane $z=2x+2y-1$ which is inside the paraboloid $z=x^2 +y^2$.
The way I proceeded was to parametrize $$ S: \begin{cases} x=1+r\cos(\theta),\\ y=1+r\sin(\theta),\\ z=3+2r(\cos(\theta)+\sin(\theta)) \end{cases} $$ where $\theta \in [0,2\pi)$ and, since the circunference projected at the XY plane is centered at (1,1), we have $r \in [\sqrt{2}-1,\sqrt{3+2(\cos(\theta)+\sin(\theta))}]$
Luckly, findind $||\vec{r}_{\theta}\times \vec{r}_r||$ wasn't so hard, and I found it to be $3r$.
The problem is the integral after that becomes painfully complicated (at least for me). I suspect there is a better solution/parametrization. Can somebody help me out?
Also, I know there is a way to do it using Stokes theorem, but I'm not suposed to use it on this one.