Answer
$\iint_D e^{(x^2+y^2)^2}dA=\int_0^12\pi re^{r^4}dr\approx 4.5951$
Work Step by Step
$D=\{(r,\theta)|0\leq r\leq 1,0\leq\theta\leq 2\pi\}$.
$\iint_De^{(x^2+y^2)^2}dA=\int_0^1\int_0^{2\pi}e^{(r^2)^2}rd\theta dr$
$=\int_0^1\int_0^{2\pi} re^{r^4}d\theta dr$
$=\int_0^12\pi re^{r^4}dr$
$\approx 4.5951$
