Answer
(\pi/4)(1-e^-4)
Work Step by Step
$\int$0 to 2 \int\0 to sqrt (4-x^2)^0.5 e^(-x^2-y^2) dydx (use values in integral signs to sketch drawing on xy plane)
0\leqTHETA\leq\pi/2 (use theta first notation to figure out bounds for theta and r)
\int0 to \pi/2 \int0 to 2 ve^{r^{2}} rdrdTHETA (new integral from theta first bound notation)
\int0 to \2 e^{-r^{2}} rdr = e^{-4})/-2+ \frac{1}{2} (solve for the inside integral and leave the outside dTHETA integral for a later step)
\int0 to \pi/2 e^{-4})/-2+ \frac{1}{2} dTHETA = (e^{-4}\pi)/-4)+(\pi/0.25) (plug answer of previous integral into the dTHETA integral and then anitderive for theta.)
= (\pi/4)(1-(e^{-4}) (Clean up the answer and this is done)
