Answer
$$\frac{{10}}{3}{y^3}$$
Work Step by Step
$$\eqalign{
& \int_y^{2y} {\left( {{x^2} + {y^2}} \right)} dx \cr
& {\text{Integrate with respect to }}x \cr
& = \left[ {\frac{1}{3}{x^3} + x{y^2}} \right]_y^{2y} \cr
& {\text{Evaluate the limits}} \cr
& = \left[ {\frac{1}{3}{{\left( {2y} \right)}^3} + \left( {2y} \right){y^2}} \right] - \left[ {\frac{1}{3}{{\left( y \right)}^3} + \left( y \right){y^2}} \right] \cr
& {\text{Simplifying}} \cr
& = \left[ {\frac{{8{y^3}}}{3} + 2{y^3}} \right] - \left[ {\frac{1}{3}{y^3} + {y^3}} \right] \cr
& = \frac{{14}}{3}{y^3} - \frac{4}{3}{y^3} \cr
& = \frac{{10}}{3}{y^3} \cr} $$
