In mathematics, in particular integral calculus, disk integration (the "disk method") is a means of calculating the volume of a solid of revolution. This makes use of the so-called "representative disk". The idea is that a "representative rectangle" (used in the most basic forms of integration -- such as ∫ x dx) can be rotated about the axis of revolution; thus generating such a disk. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... This article deals with the concept of an integral in calculus. ... A calculation is a deliberate process for transforming one or more inputs into one or more results. ... Volume (also called capacity) is a quantification of how much space an object occupies. ... In mathematics, engineering, and manufacturing, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis) that lies on the same plane. ... A synonym for ball (in geometry or topology, and in any dimension) is disk (or disc Geometry In metric geometry, a ball is a set containing all points within a specified distance of a given point. ... In geometry, a rectangle is a defined as a quadrilateral polygon in which all four angles are right angles. ... In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. ... In mathematics, engineering, and manufacturing, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis) that lies on the same plane. ...

As volume is the antiderivative of area, the integral can be used to find the volume, V, of an integrated "family" of disks. The necessary equation, for calculating such a volume, is slightly different depending on which axis is serving as the axis of revolution. These equations note that the area of a disk (one which has no height, and no volume) equals: pi (π) multiplied by the disk's squared radius (r²). The volume of a representative disk equals: πr²which is in turn multiplied by the disk's length (dx) or height (dy), that being some number approaching zero. In calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f, i. ... A family of Ouagadougou, Burkina Faso in 1997 A family is a domestic group of people, or a number of domestic groups affiliated by blood or by a variety of legal ties such as marriage, domestic partnership, adoption, surname and in some cases slavery as was the case in the... In mathematics, one often (not quite always) distinguishes between an identity, which is an assertion that two expressions are equal regardless of the values of any variables that occur within them, and an equation, which may be true for only some (or none) of the values of any such variables. ... See also the disambiguation page title equality. ... The minuscule, or lower-case, pi The mathematical constant Ï€ represents the ratio of a circles circumference to its diameter and is commonly used in mathematics, physics, and engineering. ... In its simplest form, multiplication is a quick way of adding identical numbers. ... A square as a geometric shape is described and illustrated at square (geometry). ... RADIUS (Remote Authentication Dial In User Service) is an AAA (authentication, authorization and accounting) protocol for applications such as network access or IP mobility. ... A number is an abstract entity used originally to describe quantity. ... In mathematics, the concept of a limit is used to describe the behavior of a function, as its argument gets close to either some point, or infinity; or the behavior of a sequences elements, as their index approaches infinity. ... 0 (zero) or nought is both a number and a numeral. ...

Horizontal axis of revolution

Vertical axis of revolution

For instance, consider the function f(x) = √(sin x), as it exists between x = 0 and x = π. If one imagines this function being rotated around the x-axis (so as to create a solid of revolution); then, the radius of that solid (for any value, x) is equal to √(sin x). Using the above formula, one can determine the solid to have a volume of: π ∫ [√(sin x)]² dx -- when evaluated from 0 to π. The solid has a volume of 2π. In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... A coordinate axis is one of a set of vectors that defines a coordinate system. ... Value is a term that expresses the concept of worth in general, and it is thought to be connected to reasons for certain practices, policies, or actions. ...

The "washer" method

More than one representative disk, each with a different radius and both being concentric, can be used to find the volume of more complex solids of revolution. Some mathematicians have observed that, when one deletes the area which is shared by both disks, the resulting annulus shape looks like a washer. A mathematician is a person whose area of study and research is mathematics. ... This page is about mathematics. ... Assorted washers: flat, split, star and insulated A washer is a thin disk with a hole, usually in the middle. ...

Consider the region bounded by R(x) = √x and r(x) = x². One can calculate the volume of a sphere of revolution, which has a radius of √x, as shown above -- π ∫ (√x)² dx = π / 2. One should then calculate the volume of the "inner" sphere of revolution, that having a radius of x² -- π ∫ (x²)² dx = π / 5. By subtracting the inner area, π / 2 - π / 5, one obtains the volume of the bounded area: 3π / 10. The word Boundary has a variety of meanings. ...

See also

• shell integration