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