Answer
$\dfrac{11}{3}$
Work Step by Step
We can define the domain $D$ as follows: $
D=\left\{ (x, y) | 1 \leq y \leq 2, \ y-1 \leq x \leq -3y+7 \right\}
$
Therefore, $\iint_{D} y^2 \ dA=\int_{1}^{2} \int_{y-1}^{-3y+7} y^2 \ dx \ dy \\ =\int_{1}^{2} [xy^2]_{y-1}^{-3y+7} \ dx \\= \int_1^2 (-4y^3+8y^2) \ dy \\= [\dfrac{-4y^4}{4} +\dfrac{8y^3}{3}]_1^2 \\= [-y^4+\dfrac{8y^3}{3}]_1^2 \\=-15+\dfrac{56}{3} \\=\dfrac{11}{3}$
