Evaluating $\frac{1}{2\pi} \iint_{\mathbb{R}^2} e^{\frac{-x^2}{2}} e^{\frac{-y^2}{2}} \, dA$ I'm trying to evaluate the double integral
$$
\frac{1}{2\pi} \iint_{\mathbb{R}^2} e^{\frac{-x^2}{2}} e^{\frac{-y^2}{2}} \, dA.
$$
Any ideas?

Hint: use polar coordinates:$$
\frac{1}{2\pi} \int_{\Bbb{R}^2} e^{\frac{-x^2}{2}} e^{\frac{-y^2}{2}} \, dA 
= \lim_{R\to\infty}
\frac{1}{2\pi} \int_{x^2 +y^2 \le R^2} 
e^{-\frac{x^2+y^2}{2}}  \, dA 
\\
= \lim_{R\to\infty}
\frac{1}{2\pi} \int_{0\le r \le R} 
e^{-\frac{r^2}{2}}  2\pi r dr
= \lim_{R\to\infty} \left[-e^{-\frac{r^2}{2}}
\right]_0^R
\lim_{R\to\infty} 1-e^{-\frac{R^2}{2}} = 1
$$
