View source for Method of exhaustion

The '''method of exhaustion''' (''methodus exhaustionibus'', or ''méthode des anciens'') is a method of finding the [[area]] of a [[shape]] by inscribing inside it a sequence of [[Polygon|polygons] whose areas converge to the area of the containing shape.

== Description ==

If the sequence is correctly constructed, the difference in area between the n-th polygon and the containing shape will become arbitrarily small as n becomes large.

As this difference becomes arbitrarily small, the possible values for the area of the shape are systematically "exhausted" by the lower bound areas successively established by the sequence members.

The method of exhaustion typically required a form of proof by contradiction, known as reductio ad absurdum. 

This amounts to finding an area of a region by first comparing it to the area of a second region (which can be "exhausted" so that its area becomes arbitrarily close to the true area).

The proof involves assuming that the true area is greater than the second area, and then proving that assertion false, and then assuming that it is less than the second area, and proving that assertion false, too.

== Proof by exhaustion ==

This article is about the method of finding the area of a shape using limits.

For the method of proof, see [[Proof by exhaustion]].

== See also ==

* ''[[The Method of Mechanical Theorems]]''
* ''[[The Quadrature of the Parabola]]''
* [[Trapezoidal rule]]

== External links ==

* [https://en.wikipedia.org/wiki/Method_of_exhaustion Method of exhaustion]

[[Category:Geometry]]
[[Category:Mathematics]]