State the rule for changing variables in a double integral.

Let $D$ be the region defined by

$$\begin{cases}1 / x \leq y \leq 4 x & \text { when } \frac{1}{2} \leq x \leq 1 \\ x \leq y \leq 4 / x & \text { when } 1 \leq x \leq 2\end{cases}$$

Using the transformation $u=y / x$ and $v=x y$, show that

$$\int_{D} \frac{4 x y^{3}}{x^{2}+y^{2}} d x d y=\frac{15}{2} \ln \frac{17}{2}$$