Answer
$\dfrac{4 \pi \sqrt 2}{3}$
Work Step by Step
We write the region in the polar co-ordinates as: $0 \leq r \leq \sqrt 2$ and $0 \leq \theta \leq 2\pi$
$\iint_{D} f(x,y) \ dA=\int_0^{2\pi} \int_0^{\sqrt 2} (r) (r) \ dr \ d\theta$
Now, we have:
$\int_0^{2\pi} \int_0^{\sqrt 2} (r) (r) \ dr \ d\theta=\int_0^{2\pi} [\dfrac{r^3}{3}]_0^{\sqrt 2} d\theta$
or. $=[\dfrac{2 \sqrt 2\theta}{3}]_0^{2\pi}$
or. $=\dfrac{4 \pi \sqrt 2}{3}$
