Evaluate

$$
\iint_{\mathcal{A}} e^{-x^{2}-y^{2}} d x d y,
$$

where $\mathcal{A}=\left\{(x, y) \in \mathbb{R}^{2} \mid x^{2}+y^{2} \leqslant 1\right\}$.


 Using polar coordinates, we have

$$
\begin{aligned}
\iint_{\mathcal{A}} e^{-x^{2}-y^{2}} d x d y &=\int_{0}^{2 \pi} \int_{0}^{1} \rho e^{-\rho^{2}} d \rho d \theta \\n&=-\frac{1}{2} \int_{0}^{2 \pi} \int_{0}^{1}-2 \rho e^{-\rho^{2}} d \rho d \theta \\n&=-\frac{1}{2} \int_{0}^{2 \pi}\left(e^{-1}-1\right) \\n&=\pi\left(e^{-1}-1\right) .
\end{aligned}
$$

