$$\int_{0}^a\int_{0}^\sqrt{a^2-y^2} (2x-3xy)\,dx\,dy$$
In this question I want to solve it by using polar coordinates while taking $x=r\cos{\theta}$ and $y=r\sin{\theta}$, I know that putting values of $x$ and $y$ in function and replacing $dx\,dy$ by $r\,dr\,d\theta$, but how will I determine the limits of Transformed Integral? Is there any other way without looking at the geometry of curves?
The region is just the part of $\{(x,y): x^{2}+y^{2} \leq a^{2}\}$ in the first quadrant. Hence the limts are $0<r<a$ and $0<\theta<\frac {\pi} 2$