volume under the Gaussian function?

Question

I'm trying to calculate the volume between the unit disk and the two variable analog of the Gaussian function. This is how I'm trying to calculate it: $$\int_{-1}^1\int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}e^{-x^2-y^2} dy\space dx$$ Which I've simplified into the following. $$\int_{-1}^1e^{-x^2}(\sqrt\pi erf(\sqrt{1-x^2})-\sqrt\pi erf(-\sqrt{1-x^2}))\space dx$$ Which in turn, due to the fact that $erf(x)$ is an odd function becomes: $$2\sqrt\pi\int_{-1}^1e^{-x^2}erf(\sqrt{1-x^2})\space dx$$ But after this I don't know what to do. I've tried asking this and this site but neither of them give anything but an estimation. Am I missing something? Did I mess up somewhere? Or is the answer not possible in terms of elementry functions?

Answer 1

Let your integral be $I$. Expressed in polar coordinates, $$I=\int_0^{2\pi} \int_0^1 e^{-r^2} \,r\,dr\,d\theta.$$ You should be able to take it from there.

Or, look up the chi squared distribution, and make a change of variables to bring your $e^{-x^2}$ into the form that article expects, $e^{-x^2/2}$.