Answer
$\pi a^2$
Work Step by Step
Conversion of polar coordinates and Cartesian coordinates are as follows:
a)$r^2=x^2+y^2 \implies r=\sqrt {x^2+y^2}$
b) $\tan \theta =\dfrac{y}{x} \implies \theta =\tan^{-1} (\dfrac{y}{x})$
c) $x=r \cos \theta$
d) $y=r \sin \theta$
This information suggests that the integral over a circle of $4$ quadrants is as follows:
$(2) \int_0^{a} \int_{{\frac{-\pi}{2}} }^{\frac{\pi}{2}} \int_0^{2} r dr d\theta=(2) \int_{{\frac{-\pi}{2}} }^{\frac{\pi}{2}} \dfrac{r^2}{2} d\theta$
Thus, $(2) \int_{{\frac{-\pi}{2}} }^{\frac{\pi}{2}} \dfrac{r^2}{2} d\theta=2(\dfrac{a^2}{2})\pi=\pi a^2$