I found it very hard to draw the sketch so my question is can we solve the question numerically without drawing the sketch ?
For example I have a question $$\iint_D \arctan{\frac{y}{x}}\,dxdy$$ where $D$ is domain bounded by $x^2+y^2=1$, $x^2+y^2=4$, $y=x$, $y = \sqrt{3}x$.
You will eventually have to make a physical or mental sketch of the domain to figure out the limits of integration. You can try to simplify the given equations through some kind of substitution. Polar coordinates work well here.
Keep $x^2+y^2=r^2,\theta=\arctan(y/x)$.
The domain $D$ is flanked by $r=1,r=2,\theta=\pi/4,\theta=\pi/3$. This is pretty simple to sketch.
The integral in the first quadrant becomes$$\int_{\pi/4}^{\pi/3}\int_1^2\theta rdr~d\theta$$