\displaystyle \int_{0}^{2} \left[ \dfrac{1}{2} x^2 y^2 \right]_{y=1}^{y=3} dx = \displaystyle \int_{0}^{2} \left( \dfrac{9}{2} x^2 - \dfrac{1}{2} x^2 \right) dx = \displaystyle \int_{0}^{2} 4x^2\,dx

\displaystyle = \left[ \dfrac{4}{3} x^3 \right]_{0}^{2}= \dfrac{4}{3} (8) - 0 = \dfrac{32}{3}