$\int\int xy (x – y)$ dx dy over the region bounded by xy = 4, y = 0, x = 1 and x = 4.

Let $I=\iint x y(x-1) d x d y$

Rectangular hyperbola $: x y=4 \quad$ Lines $: x=1, x=4, y=0$

Intersection of line $x=1$ and $x y=4$ is $(1,4)$ .

Intersection of line $x=4$ and $x y=4$ is $(4,1)$

$\begin{aligned} \therefore \quad & 0 \leq y \leq \frac{x}{4} \\ 1 & \leq x \leq 4 \end{aligned}$

$\begin{aligned} \therefore I &=\int_{1}^{4} \int_{0}^{\frac{x}{4}}\left(x^{2} y-x y\right) d y \ d x \\ &=\int_{1}^{4}\left[\frac{y^{2}}{2} x^{2}-\frac{y^{2} x}{2}\right]_{0}^{\frac{x}{4}} d x \end{aligned}$

$\begin{aligned} &=\int_{1}^{4}\left(8-\frac{8}{x}\right) d x \\ &=[8 x-8 \log x]_{1}^{4} \\ \therefore I &=8(3-2 \log 2) \end{aligned}$