Method of exhaustion

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This article is about the method of finding the area of a shape using limits. For the method of proof, see Proof by exhaustion.

The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area between the nth 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 idea originated with Antiphon, although it is not entirely clear how well he understood it [1]. The theory was made rigorous by Eudoxus. The first use of the term was by Gregorie de Saint-Vincent in Opus geometricum guadraturae circuli et sectionum coni in 1647.

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. This sort of proof is nonconstructive and so the answer must be known in advance.

The method of exhaustion is seen as a precursor to the methods of calculus. The development of analytical geometry and rigorous integral calculus in the 17th-19th centuries (in particular a rigorous definition of limit) subsumed the method of exhaustion so that it is no longer explicitly used to solve problems.

Archimedes used the method of exhaustion as a way to calculate π by filling the circle with a polygon of a greater and greater number of sides. The quotient formed by the area of this polygon divided by the square of the circle radius can be made arbitrarily close to the actual value of π as the number of polygon sides becomes large.

Other results he obtained with the method of exhaustion included^[1]

The area bounded by the intersection of a line and a parabola is 4/3 that of the triangle having the same base and height;
The area of an ellipse is proportional to a rectangle having sides equal to its major and minor axes;
The volume of a sphere is 4 times that of a cone having a base and height of the same radius;
The volume of a cylinder having a height equal to its diameter is 3/2 that of a sphere having the same diameter;
The area bounded by one spiral rotation and a line is 1/3 that of the circle having a radius equal to the line segment length;
Use of the method of exhaustion also led to the successful evaluation of a geometric series (for the first time).

A new form of the method of exhaustion^[2] provides a formula to evaluate the definite integral of any continuous function:

$\int\limits_a^b {f(x)\,dx = \left( {b - a} \right)} \sum\limits_{n = 1}^\infty {\sum\limits_{m = 1}^{2^n - 1} {\left( { - 1} \right)^{m + 1} } } 2^{ - n} f(a + m\left( {b - a} \right)/2^n ).$