Solid of revolution
In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis of revolution) that lies on the same plane. The surface created by this rev
Localization theorem
In mathematics, particularly in integral calculus, the localization theorem allows, under certain conditions, to infer the nullity of a function given only information about its continuity and the val
Fresnel integral
The Fresnel integrals S(x) and C(x) are two transcendental functions named after Augustin-Jean Fresnel that are used in optics and are closely related to the error function (erf). They arise in the de
Nonelementary integral
In mathematics, a nonelementary antiderivative of a given elementary function is an antiderivative (or indefinite integral) that is, itself, not an elementary function (i.e. a function constructed fro
Integral operator
An integral operator is an operator that involves integration. Special instances are:
* The operator of integration itself, denoted by the integral symbol
* Integral linear operators, which are line
Quadratic integral
In mathematics, a quadratic integral is an integral of the form It can be evaluated by completing the square in the denominator.
Time evolution of integrals
Within differential calculus, in many applications, one needs to calculate the rate of change of a volume or surface integral whose domain of integration, as well as the integrand, are functions of a
Antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function f is a differentiable function F whose derivative is equal to the origin
Indicator function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if A is a subset
Dirichlet integral
In mathematics, there are several integrals known as the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet, one of which is the improper integral of the sinc function o
Quadrature (mathematics)
In mathematics, quadrature is a historical term which means the process of determining area. This term is still used nowadays in the context of differential equations, where "solving an equation by qu
Lebesgue's decomposition theorem
In mathematics, more precisely in measure theory, Lebesgue's decomposition theorem states that for every two σ-finite signed measures and on a measurable space there exist two σ-finite signed measures
Hyperbolic sector
A hyperbolic sector is a region of the Cartesian plane bounded by a hyperbola and two rays from the origin to it. For example, the two points (a, 1/a) and (b, 1/b) on the rectangular hyperbola xy = 1,
Riemann sum
In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application
Disc integration
Disc integration, also known in integral calculus as the disc method, is a method for calculating the volume of a solid of revolution of a solid-state material when integrating along an axis "parallel
Glasser's master theorem
In integral calculus, Glasser's master theorem explains how a certain broad class of substitutions can simplify certain integrals over the whole interval from to It is applicable in cases where the in
Absolute convergence
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or com
Integration by parts operator
In mathematics, an integration by parts operator is a linear operator used to formulate integration by parts formulae; the most interesting examples of integration by parts operators occur in infinite
Surface of revolution
A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation. Examples of surfaces of revolution generated by a straight line are cyl
Radius of curvature
In differential geometry, the radius of curvature, R, is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For sur
Bioche's rules
Bioche's rules, formulated by the French mathematician (1859–1949), are rules to aid in the computation of certain indefinite integrals in which the integrand contains sines and cosines. In the follow
Conditional convergence
In mathematics, a series or integral is said to be conditionally convergent if it converges, but it does not converge absolutely.
Essential infimum and essential supremum
In mathematics, the concepts of essential infimum and essential supremum are related to the notions of infimum and supremum, but adapted to measure theory and functional analysis, where one often deal
Berezin integral
In mathematical physics, the Berezin integral, named after Felix Berezin, (also known as Grassmann integral, after Hermann Grassmann), is a way to define integration for functions of Grassmann variabl
Simpson's rule
In numerical integration, Simpson's rules are several approximations for definite integrals, named after Thomas Simpson (1710–1761). The most basic of these rules, called Simpson's 1/3 rule, or just S
Arc length
Arc length is the distance between two points along a section of a curve. Determining the length of an irregular arc segment by approximating the arc segment as connected (straight) line segments is a
Integral test for convergence
In mathematics, the integral test for convergence is a method used to test infinite series of monotonous terms for convergence. It was developed by Colin Maclaurin and Augustin-Louis Cauchy and is som
Itô calculus
Itô calculus, named after Kiyosi Itô, extends the methods of calculus to stochastic processes such as Brownian motion (see Wiener process). It has important applications in mathematical finance and st
Malliavin calculus
In probability theory and related fields, Malliavin calculus is a set of mathematical techniques and ideas that extend the mathematical field of calculus of variations from deterministic functions to
Leibniz integral rule
In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form where and the integral are functions dependen
Integration using parametric derivatives
In calculus, integration by parametric derivatives, also called parametric integration, is a method of Using known Integrals to integrate derived functions. It is often used in Physics, and is similar
Lebesgue's density theorem
In mathematics, Lebesgue's density theorem states that for any Lebesgue measurable set , the "density" of A is 0 or 1 at almost every point in . Additionally, the "density" of A is 1 at almost every p
Tonelli–Hobson test
In mathematics, the Tonelli–Hobson test gives sufficient criteria for a function ƒ on R2 to be an integrable function. It is often used to establish that Fubini's theorem may be applied to ƒ. It is na
Volume element
In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volu
Multiple integral
In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, f(x, y) or f(x, y, z). Integrals of a function o
Signed measure
In mathematics, signed measure is a generalization of the concept of (positive) measure by allowing the set function to take negative values.
Improper integral
In mathematical analysis, an improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number or positive or negative in
Limits of integration
In calculus and mathematical analysis the limits of integration (or bounds of integration) of the integral of a Riemann integrable function defined on a closed and bounded interval are the real number
Risch algorithm
In symbolic computation, the Risch algorithm is a method of indefinite integration used in some computer algebra systems to find antiderivatives. It is named after the American mathematician Robert He
Shell integration
Shell integration (the shell method in integral calculus) is a method for calculating the volume of a solid of revolution, when integrating along an axis perpendicular to the axis of revolution. This
Euler substitution
Euler substitution is a method for evaluating integrals of the form where is a rational function of and . In such cases, the integrand can be changed to a rational function by using the substitutions
Singular measure
In mathematics, two positive (or signed or complex) measures and defined on a measurable space are called singular if there exist two disjoint sets whose union is such that is zero on all measurable s
Method of exhaustion
The method of exhaustion (Latin: methodus exhaustionibus; French: méthode des anciens) is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to
Cauchy formula for repeated integration
The Cauchy formula for repeated integration, named after Augustin-Louis Cauchy, allows one to compress n antidifferentiations of a function into a single integral (cf. Cauchy's formula).
Functional integration
Functional integration is a collection of results in mathematics and physics where the domain of an integral is no longer a region of space, but a space of functions. Functional integrals arise in pro
Hyperbolic angle
In geometry, hyperbolic angle is a real number determined by the area of the corresponding hyperbolic sector of xy = 1 in Quadrant I of the Cartesian plane. The hyperbolic angle parametrises the unit
Integral of secant cubed
The integral of secant cubed is a frequent and challenging indefinite integral of elementary calculus: where is the inverse Gudermannian function, the integral of the secant function. There are a numb
Period (algebraic geometry)
In algebraic geometry, a period is a number that can be expressed as an integral of an algebraic function over an algebraic domain. Sums and products of periods remain periods, so the periods form a r
Locally integrable function
In mathematics, a locally integrable function (sometimes also called locally summable function) is a function which is integrable (so its integral is finite) on every compact subset of its domain of d
Proper integral
Proper integral is a kind of integral in Integral calculus , a branch of Mathematics in Calculus .
Integration by substitution
In calculus, integration by substitution, also known as u-substitution, reverse chain rule or change of variables, is a method for evaluating integrals and antiderivatives. It is the counterpart to th
Absolutely integrable function
In mathematics, an absolutely integrable function is a function whose absolute value is integrable, meaning that the integral of the absolute value over the whole domain is finite. For a real-valued f
Integration by parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the pro
Integration using Euler's formula
In integral calculus, Euler's formula for complex numbers may be used to evaluate integrals involving trigonometric functions. Using Euler's formula, any trigonometric function may be written in terms
Integration by reduction formulae
In integral calculus, integration by reduction formulae is method relying on recurrence relations. It is used when an expression containing an integer parameter, usually in the form of powers of eleme
Constant of integration
In calculus, the constant of integration, often denoted by , is a constant term added to an antiderivative of a function to indicate that the indefinite integral of (i.e., the set of all antiderivativ
Trigonometric substitution
In mathematics, trigonometric substitution is the replacement of trigonometric functions for other expressions. In calculus, trigonometric substitution is a technique for evaluating integrals. Moreove
Product measure
In mathematics, given two measurable spaces and measures on them, one can obtain a product measurable space and a product measure on that space. Conceptually, this is similar to defining the Cartesian
Order of integration (calculus)
In calculus, interchange of the order of integration is a methodology that transforms iterated integrals (or multiple integrals through the use of Fubini's theorem) of functions into other, hopefully
Tangent half-angle substitution
In integral calculus, the tangent half-angle substitution is a change of variables used for evaluating integrals, which converts a rational function of trigonometric functions of into an ordinary rati
Integral of the secant function
In calculus, the integral of the secant function can be evaluated using a variety of methods and there are multiple ways of expressing the antiderivative, all of which can be shown to be equivalent vi
Stochastic calculus
Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to s