\begin{align*} I^2 & = \frac{1}{2 \pi} \int_0^{2 \pi} \int_0^{\infty} e^{ -\frac{r^2}{2} } r dr d\theta\\ & = \frac{1}{2 \pi} \int_0^{2 \pi} -e^{ -\frac{r^2}{2} } \big |_0^{\infty} d\theta\\ & = \frac{1}{2 \pi} \int_0^{2 \pi} 1 \cdot d\theta\\ & = \frac{1}{2 \pi} \theta \big |_0^{2 \pi} = 1 \end{align*}