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  "body": "\n## Statement\n\nLet $E \\subseteq \\mathbb{R}^3$ be a Jordan measurable solid that lies between the planes $z = a$ and $z = b$.\nFor $z \\in [a, b]$, let $E_z = \\{(x, y) : (x, y, z) \\in E\\}$ be its cross-section at height $z$, and suppose that each $E_z$ is Jordan measurable with area $A(z)$.\nThen\n$$\\operatorname{vol}(E) = \\int_a^b A(z) \\, dz.$$\nIn particular, if two such solids have cross-sections of equal area at each height $z$, then they have equal volume.\n\n## Remarks\n\nThis follows from [[Fubini's theorem|0R0T1VWH5X]] applied to the indicator function of $E$.\nExample: the ball of radius $r$ has cross-sections of area $\\pi (r^2 - z^2)$. A cylinder of radius $r$ and height $r$ with a cone removed has the same cross-sections, which gives the volume $\\tfrac{2}{3} \\pi r^3$ for a half ball.\nThe plane version says that two regions with equal chord lengths on each line parallel to a fixed line have equal area.\nArchimedes used a similar method in the 3rd century BC. Cavalieri developed it as the method of indivisibles in 1635.\n\n## Explanations\n\n### L1 human\n\nCut two solids into thin horizontal slices.\nSuppose that each slice of one has the same area as the slice of the other at the same height. Then the two solids have the same volume, like two stacks of coins of the same size.\n",
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