mathlet here

I was trying to integrate e^(-x^2) from -n to n

I considered e^(-x^2-y^2) instead over [-n, n]^2 and did the switch to polar coordinates (x, y) -> (r, theta) calculated the jacobian
And ended up with

e^(-r^2) r dr dtheta

However, what the fuck do you do with the limits ?
Is this where math fails? I can't represent a square with polar coordinates or some shit?
Its easy to approximate with the circle inside the square and the smallest circle which contains the square but
I have never actually seen any discussion of this in any text books, or heard any professors mention this shit. It's just been informal handwaving how to get the new limits for the integral
I mean, are you even allowed to do the transformation if you can't represent the domain approriately?