Answer
$$
\iint_{D} e^{\left(x^{2}+y^{2}\right)^{2}} d A
$$
where $D$ is the disk with center at the origin and a radius of 1.
The required integral is given by:
$$
\begin{aligned}
\iint_{D} e^{\left(x^{2}+y^{2}\right)^{2}} d A & \approx 4.5951
\end{aligned}
$$
Work Step by Step
$$
\iint_{D} e^{\left(x^{2}+y^{2}\right)^{2}} d A
$$
where $D$ is the disk with center at the origin and a radius of 1. In polar coordinates, we have $x^{2}+y^{2}=r^{2} $; thus the disk D is given by:
$$ D=\left\{(r, \theta) | 0 \leq r \leq 1, \quad 0 \leq \theta \leq 2\pi \right\} $$
and, the required integral is given by:
$$
\begin{aligned}
\iint_{D} e^{\left(x^{2}+y^{2}\right)^{2}} d A &=\int_{0}^{2 \pi} \int_{0}^{1} e^{\left(r^{2}\right)^{2}} r d r d \theta \\
&=\int_{0}^{2 \pi} d \theta \int_{0}^{1} r e^{r t^{4}} d r \\
&=2 \pi \int_{0}^{1} r e^{r^{4}} d r . \text { [Using a calculator] }\\
& \approx 4.5951
\end{aligned}
$$
