This is one of my all time favourites (quoted from Problems and Theorems in Analysis by Polya and Szego).
The 3D domain $\mathcal D$ is defined by the inequalities $$-1\leq x,y,z\leq 1,\quad -\sigma\leq x,y,z\leq \sigma.$$ Show that the volume of $\mathcal D$ is $$\iiint\limits_{\mathcal D}dx\, dy\, dz=\frac{8}{\pi}\int\limits_{-\infty}^{\infty} \left(\frac{\sin t}{t}\right)^3\frac{\sin \sigma t}{t} dt.$$