$$\int_0^ {ae^{{\pi}/4}} \int_{2 \log(r/a)}^{\pi/2} f(r, \theta) \ r \ dr \ d \theta$$
In the above integral, for evaluating and sketching it, I have to change the order of integral, but I am not able to get the right approach. Is there any general method to handle change of order of integration in polar coordinates?
The answer is $\theta$ varies from $0$ to $\pi/2$ and $r$ varies from $a$ to $ae^{\theta/2}$.
I haven't ever seen a general method to handle change of order in such integrals in (between) any coordinates. In this example we have $2\ln\dfrac{r}{a}\leq \theta\leq\dfrac{\pi}{2}$ and $r\geq0$ hence $$\color{red}{0\leq r\leq ae^{\frac{\theta}{2}}}$$ and also $$0\leq r\leq ae^{\pi/4}$$ this makes an annulus and gives restriction $0\leq \dfrac{r}{a}\leq e^{\pi/4}$, taking $\log$ of sides, then $$-\infty\leq \ln\dfrac{r}{a}\leq \dfrac{\pi}{4}$$ or $-\infty\leq 2\ln\dfrac{r}{a}\leq \dfrac{\pi}{2}$ gives bound for $\theta$ $$\color{red}{-\infty\leq \theta\leq \dfrac{\pi}{2}}$$ finally your integral is $$\color{blue}{\int_{-\infty}^{\frac{\pi}{2}}\int_0^{ae^{\frac{\theta}{2}}}f(r,\theta)r \ \mathrm{dr} \ \mathrm{d\theta}}$$