Integrate e^-x^2

after you have that square region of [-1;1] for both x and y you have two integrals on x and y with limits -1 and 1. To transform that into the polar integral you have to change the limits of the integrals into something that describes that square region. You'd probably have to break the integral and in 4 parts, theta belonging to [-pi/4;pi/4], [pi/4;3pi/4], [3pi/4;5pi/4], [5pi/4;-pi/4] and have r go from zero until the the functions x=1, y=1, x=-1 and y=-1 respectively (and change those to polar coordinates).

So the limits of your integral on r would depend on theta so you wouldn't be able to separate the two integrals and you have a much more complex problem.