Let $f(x, y)$ be a function of two variables, and $R$ a region in the $x y$-plane. State the rule for evaluating $\int_{R} f(x, y) \mathrm{d} x \mathrm{~d} y$ as an integral with respect to new variables $u(x, y)$ and $v(x, y)$.

Sketch the region $R$ in the $x y$-plane defined by

$$R=\left\{(x, y): x^{2}+y^{2} \leqslant 2, x^{2}-y^{2} \geqslant 1, x \geqslant 0, y \geqslant 0\right\}$$

Sketch the corresponding region in the $u v$-plane, where

$$u=x^{2}+y^{2}, \quad v=x^{2}-y^{2}$$

Express the integral

$$I=\int_{R}\left(x^{5} y-x y^{5}\right) \exp \left(4 x^{2} y^{2}\right) \mathrm{d} x \mathrm{~d} y$$

as an integral with respect to $u$ and $v$. Hence, or otherwise, calculate $I$.