I was tasked to compute the following double integral:
$$\iint\limits_D \sqrt{\left | x-y \right |}\, dx\, dy\,,$$
where rectangular region $D$ is bounded by $0 \leq x \leq 1$ and $0 \leq y \leq 2$.
Direct integration is futile. 3D visualization of the graph reveals a trough, $z = 0$, along $y = x$. I suppose we could rotate the whole graph, including region D 45 degrees clockwise such that we can rewrite the function to integrate as $\sqrt{x}$ which will be easier to integrate.
$$\int\limits_{x=0}^1 \int\limits_{y=0}^x \sqrt{x-y}\ dy\ dx + \int\limits_{x=0}^1 \int\limits_{y=x}^2 \sqrt{y - x}\ dy\ dx = \frac{4}{15}+\frac{4}{15} \left(4 \sqrt{2}-1\right) = \frac{16 \sqrt{2}}{15}$$