Answer
$$10\sqrt 5 - 4\sqrt 2 - 14$$
Work Step by Step
$$\eqalign{
& \int_0^1 {\int_1^4 {\frac{{3y}}{{\sqrt {x + {y^2}} }}} dxdy} \cr
& {\text{Integrate with respect to }}x \cr
& = \int_0^1 {3y\left[ {2\sqrt {x + {y^2}} } \right]_1^4} dy \cr
& = 6\int_0^1 {y\left( {\sqrt {4 + {y^2}} - \sqrt {1 + {y^2}} } \right)} dy \cr
& = 3\int_0^1 {\left( {2y\sqrt {4 + {y^2}} - 2y\sqrt {1 + {y^2}} } \right)} dy \cr
& {\text{Integrating}} \cr
& = 3\left( {\frac{2}{3}} \right)\left[ {{{\left( {4 + {y^2}} \right)}^{3/2}} - {{\left( {1 + {y^2}} \right)}^{3/2}}} \right]_0^1 \cr
& = 2\left[ {{{\left( {4 + {1^2}} \right)}^{3/2}} - {{\left( {1 + {1^2}} \right)}^{3/2}}} \right] - 2\left[ {{{\left( {4 + 0} \right)}^{3/2}} - {{\left( {1 + 0} \right)}^{3/2}}} \right] \cr
& = 2\left[ {{{\left( 5 \right)}^{3/2}} - {{\left( 2 \right)}^{3/2}}} \right] - 2\left[ {8 - 1} \right] \cr
& = 2\left( {5\sqrt 5 - 2\sqrt 2 } \right) - 14 \cr
& = 10\sqrt 5 - 4\sqrt 2 - 14 \cr} $$
