Chapter 15 - Multiple Integrals - 15.9 Change of Variables in Multiple Integrals - 15.9 Exercises - Page 1100: 14

$u=xy$ and $v=\dfrac{y}{x}$

The region $R$ is defined as follows: $\dfrac{1}{x} \lt y \lt \dfrac{4}{x}, x\lt y \lt 4x$ when, $x \gt 0$ and $y \gt 0$, then we get Suppose $u=xy$ and $v=\dfrac{y}{x}$ Here, we have $\dfrac{1}{x} \lt y \lt \dfrac{4}{x}, x\lt y \lt 4x$ This implies that $1 \lt u \lt 4 \\1\lt v \lt 4$ Hence, $u=xy$ and $v=\dfrac{y}{x}$