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 creation of this process is credited to Eudoxus. Area is a physical quantity expressing the size of a part of a surface. ... In geometry, two sets have the same shape if one can be transformed to another by a combination of translations, rotations and uniform scalings. ... Converge denotes Converge PL a programming language developed by Laurence Tratt Converge, a metalcore band from Massachusetts For the mathematical meaning of this term see Convergence. ... Eudoxus of Cnidus (Greek Εύδοξος) (410 or 408 BC - 355 or 347 BC) was a Greek astronomer, mathematician, physician, scholar and friend of Plato. ...

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. Integral and differential calculus is a central branch of mathematics, developed from algebra and geometry. ... Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry, is the study of geometry using the principles of algebra. ... Integral and differential calculus is a central branch of mathematics, developed from algebra and geometry. ... A limit can be: Limit (mathematics), including: Limit of a function Limit of a sequence Limit point Net (topology) Limit (category theory) A constraint (mathematical, physical, economical, legal, etc. ...

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. Archimedes of Syracuse. ... Lower-case pi The mathematical constant Ï€ is a real number which may be defined as the ratio of a circles circumference (Greek περιφέρεια, periphery) to its diameter in Euclidean geometry, and which is in common use in mathematics, physics, and engineering. ...

Other results he obtained with the method of exhaustion included [1] (in modern language):

• 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).

The method of exhaustion typically required a form of proof by contradiction, known as reductio ad absurdum. This amounts to finding an area by first establishing upper and lower area bounds (each of which can be “exhausted” so that its area becomes arbitrarily close to the true area). The proof involves assuming that the true area equals the upper area, and then proving that assertion false, and then assuming that it equals the lower area, and proving that assertion false, too.

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

This formula can be useful when no antiderivative exists. It can also be useful for teaching integral calculus.

References

1. D.E. Smith, “History of Mathematics.” New York: Dover Publications, 1958.

2. PlanetMath: Derivation of a definite integral formula using the method of exhaustion. http://planetmath.org/encyclopedia/ExampleOfMethodOfExhaustion.html