A report on Integral, Geometry, Mathematics and Calculus

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

An illustration of Desargues' theorem, a result in Euclidean and projective geometry

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

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

Riemann–Darboux's integration (top) and Lebesgue integration (bottom)

A European and an Arab practicing geometry in the 15th century

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.

Alhazen, 11th-century Arab mathematician and physicist

A line integral sums together elements along a curve.

Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid's Elements, (c. 1310).

The quadratic formula expresses concisely the solutions of all quadratic equations

Isaac Newton developed the use of calculus in his laws of motion and gravitation.

Numerical quadrature methods: rectangle method, trapezoidal rule, Romberg's method, Gaussian quadrature

An illustration of Euclid's parallel postulate

Rubik's cube: the study of its possible moves is a concrete application of group theory

Gottfried Wilhelm Leibniz was the first to state clearly the rules of calculus.

Acute (a), obtuse (b), and straight (c) angles. The acute and obtuse angles are also known as oblique angles.

The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.

A sphere is a surface that can be defined parametrically (by or implicitly (by x2 + y2 + z2 − r2 = 0.)

Archimedes used the method of exhaustion, depicted here, to approximate the value of pi.

The logarithmic spiral of the Nautilus shell is a classical image used to depict the growth and change related to calculus.

Visual checking of the Pythagorean theorem for the (3, 4, 5) triangle as in the Zhoubi Suanjing 500–200 BC. The Pythagorean theorem is a consequence of the Euclidean metric.

The numerals used in the Bakhshali manuscript, dated between the 2nd century BC and the 2nd century AD.

$The Koch snowflake, with fractal dimension=log4/log3 and topological dimension=1$

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.

Differential geometry uses tools from calculus to study problems involving curvature.

Leonhard Euler created and popularized much of the mathematical notation used today.

Carl Friedrich Gauss, known as the prince of mathematicians

Discrete geometry includes the study of various sphere packings.

The Cayley graph of the free group on two generators a and b

Bou Inania Madrasa, Fes, Morocco, zellige mosaic tiles forming elaborate geometric tessellations

The Pythagoreans discovered that the sides of a triangle could have incommensurable lengths.

Euler's identity, which American physicist Richard Feynman once called "the most remarkable formula in mathematics".

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

- Integral

Geometry is, with arithmetic, one of the oldest branches of mathematics.

- Geometry

Mathematics is an area of knowledge that includes such topics as numbers (arithmetic, number theory), formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).

- Mathematics

Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the 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.

- 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.

- Calculus

Along with differentiation, integration is a fundamental, essential operation of calculus, and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others.

- Integral

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).

- Mathematics

This was a necessary precursor to the development of calculus and a precise quantitative science of physics.

- Geometry

In calculus, area and volume can be defined in terms of integrals, such as the Riemann integral or the Lebesgue integral.

- Geometry

Area can sometimes be found via geometrical compass-and-straightedge constructions of an equivalent square.

- Integral

1 related topic with Alpha

Area

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Quantity that expresses the extent of a region on the plane or on a curved surface.

This square and this disk both have the same area (see: squaring the circle).

A square metre quadrat made of PVC pipe.

Although there are 10 mm in 1 cm, there are 100 mm2 in 1 cm2.

A diagram showing how a parallelogram can be re-arranged into the shape of a rectangle.

A parallelogram split into two equal triangles.

A circle can be divided into sectors which rearrange to form an approximate parallelogram.

Archimedes showed that the surface area of a sphere is exactly four times the area of a flat disk of the same radius, and the volume enclosed by the sphere is exactly 2/3 of the volume of a cylinder of the same height and radius.

Integration can be thought of as measuring the area under a curve, defined by f(x), between two points (here a and b).

The area between two graphs can be evaluated by calculating the difference between the integrals of the two functions

In mathematics, the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number.

For shapes with curved boundary, calculus is usually required to compute the area.

In addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry.

The development of integral calculus in the late 17th century provided tools that could subsequently be used for computing more complicated areas, such as the area of an ellipse and the surface areas of various curved three-dimensional objects.