Answer
$\frac{\pi}{4}[1-e^{-4}]$
Work Step by Step
$\int_0^2\int_0^{\sqrt {4-x^{2}}}e^{-x^{2}-y^{2}}dydx$
$y$ = $\sqrt {4-x^{2}}$
$y^2$ = $4-x^{2}$
$x^{2}+y^2$ = $4$
$r^2$ = $4$
$r$ = $2$
$x$ = $2$
$r\cosθ$ =$2$
$2\cosθ$ =$2$
$\cosθ$ = $1$
$θ$ = $0$
$x$ = $0$
$r\cosθ$ =$0$
$2\cosθ$ =$0$
$\cosθ$ = $0$
$θ$ = $\frac{\pi}{2}$
= $\int_0^{\frac{\pi}{2}}\int_0^2{e^{-r^2}}rdrdθ$
= $\frac{1}{2}\int_0^{\frac{\pi}{2}}{-e^{-r^2}} |_0^2dθ$
= $\frac{1}{2}\int_0^{\frac{\pi}{2}} [1-e^{-4}]dθ$
= $\frac{1}{2}[1-e^{-4}][θ]_0^{\frac{\pi}{2}}$
= $\frac{\pi}{4}[1-e^{-4}]$
