A report on Integral

A definite integral of a function can be represented as the signed area of the region bounded by its graph.
Riemann–Darboux's integration (top) and Lebesgue integration (bottom)
A line integral sums together elements along a curve.
Numerical quadrature methods: rectangle method, trapezoidal rule, Romberg's method, Gaussian quadrature

Integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data.

- Integral
A definite integral of a function can be represented as the signed area of the region bounded by its graph.

68 related topics with Alpha

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Archimedes used the method of exhaustion to calculate the area under a parabola.

Calculus

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Mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.

Mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.

Archimedes used the method of exhaustion to calculate the area under a parabola.
Alhazen, 11th-century Arab mathematician and physicist
Isaac Newton developed the use of calculus in his laws of motion and gravitation.
Gottfried Wilhelm Leibniz was the first to state clearly the rules of calculus.
Maria Gaetana Agnesi
The logarithmic spiral of the Nautilus shell is a classical image used to depict the growth and change related to calculus.

It has two major branches, differential calculus and integral calculus; differential calculus concerns instantaneous rates of change, and the slopes of curves, while integral calculus concerns accumulation of quantities, and areas under or between curves.

The slope field of, showing three of the infinitely many solutions that can be produced by varying the arbitrary constant c.

Antiderivative

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Antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function

Antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function

The slope field of, showing three of the infinitely many solutions that can be produced by varying the arbitrary constant c.

Antiderivatives are related to definite integrals through the second fundamental theorem of calculus: the definite integral of a function over a closed interval where the function is Riemann integrable is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval.

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Infinitesimal

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Infinitesimal or infinitesimal number is a quantity that is closer to 0|zero than any standard real number, but that is not zero.

Infinitesimal or infinitesimal number is a quantity that is closer to 0|zero than any standard real number, but that is not zero.

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An infinite number of infinitesimals are summed to calculate an integral.

The graph of a function, drawn in black, and a tangent line to that graph, drawn in red. The slope of the tangent line is equal to the derivative of the function at the marked point.

Derivative

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In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).

The graph of a function, drawn in black, and a tangent line to that graph, drawn in red. The slope of the tangent line is equal to the derivative of the function at the marked point.
This function does not have a derivative at the marked point, as the function is not continuous there (specifically, it has a jump discontinuity).

The fundamental theorem of calculus relates antidifferentiation with integration.

3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)

Mathematics

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Area of knowledge that includes such topics as numbers , formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).

Area of knowledge that includes such topics as numbers , formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).

3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)
The distribution of prime numbers is a central point of study in number theory. This Ulam spiral serves to illustrate it, hinting, in particular, at the conditional independence between being prime and being a value of certain quadratic polynomials.
The quadratic formula expresses concisely the solutions of all quadratic equations
Rubik's cube: the study of its possible moves is a concrete application of group theory
The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.
Archimedes used the method of exhaustion, depicted here, to approximate the value of pi.
The numerals used in the Bakhshali manuscript, dated between the 2nd century BC and the 2nd century AD.
A page from al-Khwārizmī's Algebra
Leonardo Fibonacci, the Italian mathematician who introduced the Hindu–Arabic numeral system invented between the 1st and 4th centuries by Indian mathematicians, to the Western World.
Leonhard Euler created and popularized much of the mathematical notation used today.
Carl Friedrich Gauss, known as the prince of mathematicians
The front side of the Fields Medal
Euler's identity, which American physicist Richard Feynman once called "the most remarkable formula in mathematics".

Calculus, consisting of the two subfields infinitesimal calculus and integral calculus, is the study of continuous functions, which model the typically nonlinear relationships between varying quantities (variables).

Archimedes Thoughtful by Domenico Fetti (1620)

Archimedes

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Approximation of pi; defining and investigating the spiral that now bears his name; and devising a system using exponentiation for expressing very large numbers.

Approximation of pi; defining and investigating the spiral that now bears his name; and devising a system using exponentiation for expressing very large numbers.

Archimedes Thoughtful by Domenico Fetti (1620)
The Death of Archimedes (1815) by Thomas Degeorge
Cicero Discovering the Tomb of Archimedes (1805) by Benjamin West
The Archimedes' screw can raise water efficiently.
Archimedes may have used mirrors acting collectively as a parabolic reflector to burn ships attacking Syracuse.
Archimedes calculates the side of the 12-gon from that of the hexagon and for each subsequent doubling of the sides of the regular polygon.
A proof that the area of the parabolic segment in the upper figure is equal to 4/3 that of the inscribed triangle in the lower figure from Quadrature of the Parabola.
Frontpage of Archimedes' Opera, in Greek and Latin, edited by David Rivault (1615).
A sphere has 2/3 the volume and surface area of its circumscribing cylinder including its bases.
Ostomachion is a dissection puzzle found in the Archimedes Palimpsest.
In 1906, the Archimedes Palimpsest revealed works by Archimedes thought to have been lost.
The Fields Medal carries a portrait of Archimedes.
Artistic interpretation of Archimedes' mirror used to burn Roman ships. Painting by Giulio Parigi, c. 1599.
Bronze statue of Archimedes in Berlin

Archimedes was able to use indivisibles (a precursor to infinitesimals) in a way that is similar to modern integral calculus.

Gregory of Saint Vincent

Method of exhaustion

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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.

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.

Gregory of Saint Vincent
Archimedes used the method of exhaustion to compute the area inside a circle

The development of analytical geometry and rigorous integral calculus in the 17th-19th centuries subsumed the method of exhaustion so that it is no longer explicitly used to solve problems.

Two stacks of coins with the same volume, illustrating Cavalieri's principle in three dimensions

Cavalieri's principle

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As follows:

As follows:

Two stacks of coins with the same volume, illustrating Cavalieri's principle in three dimensions
Bonaventura Cavalieri, the mathematician the principle is named after.
The disk-shaped cross-section of the sphere has the same area as the ring-shaped cross-section of that part of the cylinder that lies outside the cone.
If a hole of height h is drilled straight through the center of a sphere, the volume of the remaining band does not depend on the size of the sphere. For a larger sphere, the band will be thinner but longer.
The horizontal cross-section of the region bounded by two cycloidal arcs traced by a point on the same circle rolling in one case clockwise on the line below it, and in the other counterclockwise on the line above it, has the same length as the corresponding horizontal cross-section of the circle.

Today Cavalieri's principle is seen as an early step towards integral calculus, and while it is used in some forms, such as its generalization in Fubini's theorem, results using Cavalieri's principle can often be shown more directly via integration.

A measuring cup can be used to measure volumes of liquids. This cup measures volume in units of cups, fluid ounces, and millilitres.

Volume

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Scalar quantity expressing the amount of three-dimensional space enclosed by a closed surface.

Scalar quantity expressing the amount of three-dimensional space enclosed by a closed surface.

A measuring cup can be used to measure volumes of liquids. This cup measures volume in units of cups, fluid ounces, and millilitres.
A cone, sphere and cylinder of radius r and height h

Volumes of complicated shapes can be calculated with integral calculus if a formula exists for the shape's boundary.

Fundamental theorem of calculus (animation)

Fundamental theorem of calculus

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Fundamental theorem of calculus (animation)

The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating the gradient) with the concept of integrating a function (calculating the area under the curve).