Volume
Elementary
Think of a solid object, like a box or a ball. The volume of the solid is the amount of space it takes up, measured in cubic units.
Undergraduate
Consider a region $R \subseteq \reals^n$. The volume of $R$ is its three-dimensional (or $n$-dimensional) size:
\[\operatorname{vol}(R) = \int_R 1 \mathrm{d}x.\]