integration - Double polar integral $int_{-infty}^{infty}int_{-infty}^{infty} e^{-frac{x^2+y^2}2},dx,dy$

Evaluate $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} e^{-\frac{x^2+y^2}2}\,dx\,dy$$ using polar coordinates, where the upper limits of the both integrals are infinity and their lower limits are -infinity.

the only reason why I am confused by these is the limits of the integrals. I can't visualise the area of integration and I do not know how to convert them into polar limits. I know that the integrand will be $re^{-\frac{r^2}2}$.

Answer

By polar coordinates, since we are integrating over all the $x-y$ plane, we have that

$$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} e^{-\frac{x^2+y^2}2}\,dx\,dy=\int_0^{2\pi}\int_0^{\infty}re^{-\frac{r^2}2}\,dr\, d\theta$$

then note that $\frac{d}{dr}\left(e^{-\frac{r^2}2}\right)=-re^{-\frac{r^2}2}$ and use that

$$\int_0^{2\pi}\int_0^{\infty}re^{-\frac{r^2}2}\,dr\, d\theta =\lim_{R\to \infty} \int_0^{2\pi}\int_0^{R}re^{-\frac{r^2}2}\,dr\, d\theta$$