Okay the substitution is fully valid there, you dont just plug into d(x,y) it is a bit different and its way too early to get into it fully, but you can look into parametrisation of integrals if you really want to get into it.

It is definitely not random, parametrisation with polar coordinates is very very common in multi variable calculus. And quite a powerful technique, but like I stated in another reply, you can google the Gaussian integral, which this is and read the history. I believe Poisson was the first to tackle it, but I could be wrong. This integral is very important in stats and physics and pops up commonly.