\int_{1}^{\frac{1}{2}}\int_{1}^{\frac{1}{2}}\frac{x^{2}+y^{2}}{xy}\text{d}x\text{d}y = \int_{1}^{\frac{1}{2}}\int_{1}^{\frac{1}{2}}\frac{x^2}{xy}+\frac{y^2}{xy}\text{d}x\text{d}y = \int_{1}^{\frac{1}{2}}\left[\frac{x^3}{3xy}+\frac{xy^2}{2xy}\right]_{1}^{\frac{1}{2}} = \int_{1}^{\frac{1}{2}}\left[-\frac{1}{4y}\right]\text{d}y= \left[-\frac{1}{4y}\right]_{1}^{\frac{1}{2}}= \mathbf{\frac{1}{8}}