[Calculus] Gaussian integral derivation. How come integral from -inf to inf dxdy switches to 0 to 2pi and 0 to inf when switching to (r,theta)?

I know about the Jacobian and the transformation from dxdy to rdrd(theta) but how do you know to switch the bounds of the integral besides "well r and theta can't be negative"

follow up question: how come integral of e^(-x^2) dx squared is equal to e^(-x^2) e^(-y^2) dx dy I've tried to prove that integral e^x dx is equal to e^u d(u) but that only works with x=u