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
$$\iiint\limits_{\mathcal D}dx\, dy\, dz=\frac{8}{\pi}\int\limits_{-\infty}^{\infty}
\left(\frac{\sin t}{t}\right)^3\frac{\sin \sigma t}{t} dt.$$

Hint. Show first that the number of integer lattice points that satisfy the conditions
$$-n\leq x,y,z\leq n,\quad -s\leq x+y+z\leq s$$
for some $n$, $s\in\mathbb N$, is equal to
$$\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\left(\frac{\sin \frac{2n+1}{2}t}{sin\frac {t}{2}}\right)^3\frac{\sin \frac{2s+1}{2}t}{\sin \frac{t}{2}} dt. $$

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**Edit.** Check out the [lecture notes][1] by Oliver Knill for some classical examples of volume computations and  fabulous illustrations.






  [1]: http://abel.math.harvard.edu/~knill/teaching/math21a/s.pdf