Recall the method for evaluating a double integral, $f(x,y)dA,$ over a rectangular region $R.$ Let $R$ be the rectangle defined by the inequalities $axb$ and $cyd.$ Then the desired integral can be written as below.

${}_{R}f(x,y)dA={\displaystyle \underset{C}{\overset{d}{}}}[{\displaystyle \underset{a}{\overset{b}{}}}f(x,y)dx]dy$

We are given the function $f(x,y)=x+6y.$ Therefore, the integral we are interested in$,$ over the rectangular region $1x2$ and $0y1$ is as follows.

$[$:${\displaystyle \underset{0}{\overset{1}{}}}[{\displaystyle \underset{1}{\overset{2}{}}}(\frac{3}{2}+6y{)}_x)dx$