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