Answer
$$\frac{{117}}{2}$$
Work Step by Step
$$\eqalign{
& \iint\limits_R {\left( {x + 2y} \right)}dA;\,\,\,\,\,\,\,R = \left\{ {\left( {x,y} \right):0 \leqslant x \leqslant 3,\,\,\,\,1 \leqslant y \leqslant 4} \right\} \cr
& {\text{Convert to an iterated integral substituting the region }}R \cr
& = \int_0^3 {\int_1^4 {\left( {x + 2y} \right)} } dydx \cr
& = \int_1^3 {\left[ {\int_1^4 {\left( {x + 2y} \right)} dy} \right]} dx \cr
& {\text{solve the inner integral}}{\text{, treat }}x{\text{ as a constant}} \cr
& \int_1^4 {\left( {x + 2y} \right)} dy \cr
& = \left[ {xy + {y^2}} \right]_1^4 \cr
& {\text{evaluating the limits for the variable }}y \cr
& = \left[ {x\left( 4 \right) + {{\left( 4 \right)}^2}} \right] - \left[ {x\left( 1 \right) + {{\left( 1 \right)}^2}} \right] \cr
& {\text{simplifying}} \cr
& = 4x + 16 - x - 1 \cr
& = 3x + 15 \cr
& \cr
& = \int_0^3 {\left[ {\int_1^4 {\left( {x + 2y} \right)} dy} \right]} dx = \int_0^3 {\left( {3x + 15} \right)} dx \cr
& {\text{integrating}} \cr
& = \left( {\frac{{3{x^2}}}{2} + 15x} \right)_0^3 \cr
& {\text{evaluate}} \cr
& = \left( {\frac{{3{{\left( 3 \right)}^2}}}{2} + 15\left( 3 \right)} \right) - \left( {\frac{{3{{\left( 0 \right)}^2}}}{2} + 15\left( 0 \right)} \right) \cr
& = \frac{{27}}{2} + 45 \cr
& = \frac{{117}}{2} \cr} $$
