Answer
$$110$$
Work Step by Step
$$\eqalign{
& \iint\limits_R {\left( {{x^2} + 2{y^2}} \right)}dxdy;\,\,\,\,\,\,\,\,\,\,\,\,\,\,0 \leqslant x \leqslant 5,\,\,\,\,\,\,\,0 \leqslant y \leqslant 2 \cr
& {\text{Replacing the limits for the region }}R \cr
& = \int_0^2 {\int_0^5 {\left( {{x^2} + 2{y^2}} \right)dx} dy} \cr
& = \int_0^2 {\left[ {\int_0^5 {\left( {{x^2} + 2{y^2}} \right)dx} } \right]dy} \cr
& {\text{solve the inner integral treat }}y{\text{ as a constant}} \cr
& = \int_0^5 {\left( {{x^2} + 2{y^2}} \right)dx} \cr
& {\text{integrate by using the power rule}} \cr
& = \left( {\frac{{{x^3}}}{3} + 2x{y^2}} \right)_0^5 \cr
& {\text{evaluating the limits in the variable }}x \cr
& = \left( {\frac{{{{\left( 5 \right)}^3}}}{3} + 2\left( 5 \right){y^2}} \right) - \left( {\frac{{{{\left( 0 \right)}^3}}}{3} + 2\left( 0 \right){y^2}} \right) \cr
& {\text{simplifying}} \cr
& = \frac{{125}}{3} + 10{y^2} \cr
& \cr
& \int_0^2 {\left[ {\int_0^5 {\left( {{x^2} + 2{y^2}} \right)dx} } \right]dy} = \int_0^2 {\left( {\frac{{125}}{3} + 10{y^2}} \right)dy} \cr
& {\text{integrating}} \cr
& = \left( {\frac{{125}}{3}y + \frac{{10{y^3}}}{3}} \right)_0^2 \cr
& {\text{evaluate}} \cr
& = \left( {\frac{{125}}{3}\left( 2 \right) + \frac{{10{{\left( 2 \right)}^3}}}{3}} \right) - \left( {\frac{{125}}{3}\left( 0 \right) + \frac{{10{{\left( 0 \right)}^3}}}{3}} \right) \cr
& {\text{simplifying}} \cr
& = 110 - 0 \cr
& = 110 \cr} $$