$\int_0^{\frac{\pi}{4}} \int_{0}^{\frac{\sqrt{2}}{2} \csc{\theta}} f(r \cos(\theta), r \sin(\theta)) r dr d\theta$ is wrong?
Solution 1:
Plotting the original region reveals it to be merely a circular sector with radius $1$ bounded by $\theta=0,\pi/4$, so the correct bounds are $$\int_0^{\pi/4}\int_0^1(\dots)\,dr\,d\theta$$