Why is $-\sqrt{\pi}$ not an answer to the Gaussian Integral?

Question

The polar-coordinate proof for the Gaussian integral ends up with the integral squared equaling $\pi$. From there, I understand that $\sqrt{\pi}$ is the solution to the integral, but why isn't $-\sqrt{\pi}$ also a solution? Is that just not how polar coordinates work?

Answer 1

The integral of a positive function is positive and $x\mapsto e^{-x^2}$ is certainly positive.