Cavalieri's principle

Before computing volumes via the integral, let us look at an intuitive principle that characterises them geometrically, without yet using integral calculus: Cavalieri’s principle (Bonaventura Cavalieri, a pupil of Galileo, 1635; Boyer, A History of Mathematics, ch. 14). It is the same principle that justifies, for instance, the equidecomposability of plane figures seen in the proof of Pythagoras’s theorem.

Theorem — Cavalieri's principle, 2D version

Given two plane figures $A, B$ contained between two parallel lines $r$ and $s$: if every line parallel to $r$ cuts $A$ and $B$ in segments of equal length, then $A$ and $B$ have the same area.

Theorem — Cavalieri's principle, 3D version

Given two solids $S_1, S_2$ contained between two parallel planes $\pi_1$ and $\pi_2$: if every plane parallel to $\pi_1$ cuts $S_1$ and $S_2$ in cross-sections of equal area, then $S_1$ and $S_2$ have the same volume.

Right cylinder and oblique cylinder: at every height the cross-sections have the same area, so by Cavalieri they have the same volume.

Remark — From the principle to the integral

Cavalieri himself, summing “infinitely many infinitesimal cross-sections”, anticipated integral calculus by a century. The formula $V = \int_a^b A(z)\,dz$ is the modern version of his principle: the volume is the accumulation along $z$ of the areas $A(z)$ of the cross-sections. Cavalieri stated it without the symbol $\int$; we prove it with the integral in the following section.

Links

Topics: Integral
Concepts: Definite integral · Cavalieri’s principle · Volume by cross-sections
Methods: Cavalieri’s principle
Skills: Proving · Synthetic geometry
People: Bonaventura Cavalieri · Galilei