Cavalieri's principle

Statement

Let $E \subseteq \mathbb{R}^3$ be a Jordan measurable solid that lies between the planes $z = a$ and $z = b$. For $z \in [a, b]$, 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 $A(z)$. Then $$\operatorname{vol}(E) = \int_a^b A(z) \, dz.$$ In particular, if two such solids have cross-sections of equal area at each height $z$, then they have equal volume.

Remarks

This follows from Fubini's theorem applied to the indicator function of $E$. Example: 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. The plane version says that two regions with equal chord lengths on each line parallel to a fixed line have equal area. Archimedes used a similar method in the 3rd century BC. Cavalieri developed it as the method of indivisibles in 1635.

Explanations

L1 human

Cut two solids into thin horizontal slices. Suppose 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.