Answer
The total charge on the rectangle $D$ is $285 \mathrm{C} $
Work Step by Step
The total charge $ Q$ is given by
$$
\begin{aligned} Q & =\iint_{D} \sigma(x, y) d A \\
&=\int_{0}^{5} \int_{2}^{5}(2 x+4 y) d y d x \\
& =\int_{0}^{5}\left[2 x y+2 y^{2}\right]_{y=2}^{y=5} d x \\
&=\int_{0}^{5}(10 x+50-4 x-8) d x=\int_{0}^{5}(6 x+42) d x \\
&=\left[3 x^{2}+42 x\right]_{0}^{5} \\
&=75+210 \\
&=285 \mathrm{C} \end{aligned}
$$
Thus, the total charge is $ 285 \mathrm{C} $