{
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  "body": "\n## Statement\n\nLet $a_i \u003c b_i$ for $i = 1, \\dots, n$, and let $Q = [a_1, b_1] \\times \\dots \\times [a_n, b_n] \\subseteq \\mathbb{R}^n$.\nThen $Q$ is Jordan measurable, and its volume is\n$$\\operatorname{vol}(Q) = \\int_Q 1 \\, dV = (b_1 - a_1)(b_2 - a_2) \\cdots (b_n - a_n).$$\n\n## Remarks\n\nFor a box, the [[Jordan measure|0565DC8RMK]] and the Riemann integral of the constant function $1$ give the same value.\nThe integral can be computed with [[Fubini's theorem|0R0T1VWH5X]] as an iterated integral of $1$, one factor $b_i - a_i$ at a time.\nOpen and half-open boxes with the same edges have the same volume, because the faces of $Q$ have volume zero.\nFor $n = 3$ 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|0GTBGMZXV8]] $\\iiint_E 1 \\, dV$.\n\n## Explanations\n\n### L1 human\n\nThe volume of a box is the product of its side lengths.\nIn the theory of multiple integrals this is the starting point: Riemann sums and Jordan measure are both built from volumes of small boxes.\n",
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