Volume of a box

Statement

Let $a_i < b_i$ for $i = 1, \dots, n$, and let $Q = [a_1, b_1] \times \dots \times [a_n, b_n] \subseteq \mathbb{R}^n$. Then $Q$ is Jordan measurable, and its volume is $$\operatorname{vol}(Q) = \int_Q 1 \, dV = (b_1 - a_1)(b_2 - a_2) \cdots (b_n - a_n).$$

Remarks

For a box, the Jordan measure and the Riemann integral of the constant function $1$ give the same value. The integral can be computed with Fubini's theorem as an iterated integral of $1$, one factor $b_i - a_i$ at a time. Open and half-open boxes with the same edges have the same volume, because the faces of $Q$ have volume zero. For $n = 3$ this is the formula length times width times height, a case of base area times height. The volume of a general solid $E$ is the triple integral $\iiint_E 1 \, dV$.

Explanations

L1 human

The volume of a box is the product of its side lengths. In the theory of multiple integrals this is the starting point: Riemann sums and Jordan measure are both built from volumes of small boxes.