Answer
$ =e^{-1}-4e^{-4}\approx 0.295$
Work Step by Step
$\int ^{4}_{1}\int ^{\sqrt {x}}_{1}2ye^{-x}dydx=\int ^{4}_{1}2\times \dfrac {1}{2}y^{2}]^{\sqrt {x}}_{1}e^{-x}dx=\int ^{4}_{1}\left( x-1\right) e^{-x}dx=-xe^{-x}]^{4}_{1}=-4e^{-4}-\left( -1\times e^{-1}\right) =e^{-1}-4e^{-4}\approx 0.295 $
