Answer
$${e^4} - {e^2} - 2$$
Work Step by Step
$$\eqalign{
& \int_1^4 {\int_0^2 {{e^{y\sqrt x }}dydx} } \cr
& {\text{Integrate with respect to }}y \cr
& = \int_1^4 {\left[ {\frac{1}{{\sqrt x }}{e^{y\sqrt x }}} \right]_0^2} dx \cr
& {\text{Evaluate the limits}} \cr
& = \int_1^4 {\frac{1}{{\sqrt x }}\left[ {{e^{2\sqrt x }} - {e^0}} \right]} dx \cr
& = \int_1^4 {\frac{1}{{\sqrt x }}\left[ {{e^{2\sqrt x }} - 1} \right]} dx \cr
& = \int_1^4 {\left( {\frac{1}{{\sqrt x }}{e^{2\sqrt x }} - {x^{ - 1/2}}} \right)} dx \cr
& {\text{Integrating}} \cr
& = \left[ {{e^{2\sqrt x }} - 2\sqrt x } \right]_1^4 \cr
& = \left( {{e^{2\sqrt 4 }} - 2\sqrt 4 } \right) - \left( {{e^{2\sqrt 1 }} - 2\sqrt 1 } \right) \cr
& {\text{Simplifying}} \cr
& = \left( {{e^4} - 4} \right) - \left( {{e^2} - 2} \right) \cr
& = {e^4} - 4 - {e^2} + 2 \cr
& = {e^4} - {e^2} - 2 \cr} $$
