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

Question

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 1

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

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

Answer 2

You want to integrate over the entire space $\mathbb{R}^{2}$ so, when you consider the polar coordinates $x = r\cos\theta$ and $y=r\sin\theta$, the natural way to "fill the entire space" is to take $0 \le \theta \le 2\pi$ and $0\le r \le +\infty$. Furthermore, note that $$re^{-\frac{r^{2}}{2}} = -\frac{d}{dr}e^{\frac{r^{2}}{2}}$$