Answer
$32 \pi \sqrt 3$
Work Step by Step
Here we have $dS=\iint_D\sqrt {1+(1)^2 +(1)^2} dy dx=\iint_D\sqrt {3} dy dx$
and $\iint_S x dS=\iint_D (x^2+y^2) \sqrt {4+x+y}\sqrt 3 dx dy$
or, $\int_0^{2\pi}\int_0^{2}r^2(4+r \cos \theta+r \sin \theta)(\sqrt 3)(r dr) d\theta=(\sqrt 3) \int_0^{2\pi}\int_0^2 4r^3+r^4\cos \theta+r^4 \sin \theta dr d \theta$
or, $=(\sqrt 3) \int_0^{2 \pi} [r^4+\dfrac{r^5(\cos \theta+\sin \theta)}{5}]_0^2 d\theta$
or, $=\sqrt 3 [16 \theta+\dfrac{32(\sin \theta-\cos \theta)}{5}]_0^{2 \pi}$
or, $=32 \pi (\sqrt 3)$
