How to prove that the volume of a solid $S \subset \mathbb{R}^3$ is $\iiint\limits_S \;\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z.$

Question

How to prove that the volume of a solid $S \subset \mathbb{R}^3$ is

$$\iiint\limits_S \;\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z.$$