Answer
$$\sin \left( 4 \right) - 4\cos \left( 4 \right)$$
Work Step by Step
$$\eqalign{
& \int_0^2 {\int_{{y^2}}^4 {\sqrt x \sin x} dxdy} \cr
& {\text{Switch the order of integration using the region shown below}} \cr
& \int_0^2 {\int_{{y^2}}^4 {\sqrt x \sin x} dxdy} = \int_0^4 {\int_0^{\sqrt x } {\sqrt x \sin x} dydx} \cr
& {\text{Integrating}} \cr
& = \int_0^4 {\left[ {y\sqrt x \sin x} \right]_0^{\sqrt x }dx} \cr
& = \int_0^4 {\left[ {\sqrt x \sqrt x \sin x - \left( 0 \right)\sqrt x \sin x} \right]dx} \cr
& = \int_0^4 {x\sin xdx} \cr
& {\text{Integrating by parts we obtain}} \cr
& = \left[ {\sin x - x\cos x} \right]_0^4 \cr
& = \left[ {\sin \left( 4 \right) - \left( 4 \right)\cos \left( 4 \right)} \right] - \left[ {2\sin \left( 0 \right) - 2\left( 0 \right)\cos \left( 0 \right)} \right] \cr
& = \sin \left( 4 \right) - 4\cos \left( 4 \right) \cr} $$
