Let $A$ be the closed planar region given by

$$y \leqslant x \leqslant 2 y, \quad \frac{1}{y} \leqslant x \leqslant \frac{2}{y} .$$

(i) Evaluate by means of a suitable change of variables the integral

$$\int_{A} \frac{x}{y} d x d y$$

(ii) Let $C$ be the boundary of $A$. Evaluate the line integral

$$\oint_{C} \frac{x^{2}}{2 y} d y-d x$$

by integrating along each section of the boundary.

(iii) Comment on your results.