A report on MathematicsGeometry and Integral

3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)
An illustration of Desargues' theorem, a result in Euclidean and projective geometry
A definite integral of a function can be represented as the signed area of the region bounded by its graph.
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.
A European and an Arab practicing geometry in the 15th century
Riemann–Darboux's integration (top) and Lebesgue integration (bottom)
The quadratic formula expresses concisely the solutions of all quadratic equations
Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid's Elements, (c. 1310).
A line integral sums together elements along a curve.
Rubik's cube: the study of its possible moves is a concrete application of group theory
An illustration of Euclid's parallel postulate
Numerical quadrature methods: rectangle method, trapezoidal rule, Romberg's method, Gaussian quadrature
The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.
Acute (a), obtuse (b), and straight (c) angles. The acute and obtuse angles are also known as oblique angles.
Archimedes used the method of exhaustion, depicted here, to approximate the value of pi.
A sphere is a surface that can be defined parametrically (by  or implicitly (by x2 + y2 + z2 − r2 = 0.)
The numerals used in the Bakhshali manuscript, dated between the 2nd century BC and the 2nd century AD.
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.
A page from al-Khwārizmī's Algebra
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.
A tiling of the hyperbolic plane
Leonhard Euler created and popularized much of the mathematical notation used today.
Differential geometry uses tools from calculus to study problems involving curvature.
Carl Friedrich Gauss, known as the prince of mathematicians
A thickening of the trefoil knot
The front side of the Fields Medal
Quintic Calabi–Yau threefold
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".

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

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

- Geometry

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

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

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

4 related topics with Alpha

Overall

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

Calculus

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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.

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.

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 combined area of these three shapes is approximately 15.57 squares.

Area

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

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

The combined area of these three shapes is approximately 15.57 squares.
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.
The area of this rectangle is lw.
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.

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.

An illustration of Euclid's proof of the Pythagorean theorem.

Greek mathematics

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An illustration of Euclid's proof of the Pythagorean theorem.
Detail of Pythagoras with a tablet of ratios, from The School of Athens by Raphael. Vatican Palace, Rome, 1509.
A fragment from Euclid's Elements (c. 300 BC), widely considered the most influential mathematics textbook of all time.
Cover of Arithmetica written by Greek Mathematician Diophantus

Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean.

Greek mathematics constitutes an important period in the history of mathematics: fundamental in respect of geometry and for the idea of formal proof.

Greek mathematicians also contributed to number theory, mathematical astronomy, combinatorics, mathematical physics, and, at times, approached ideas close to the integral calculus.

Informally, a measure has the property of being monotone in the sense that if A is a subset of B, the measure of A is less than or equal to the measure of B. Furthermore, the measure of the empty set is required to be 0.

Measure (mathematics)

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Informally, a measure has the property of being monotone in the sense that if A is a subset of B, the measure of A is less than or equal to the measure of B. Furthermore, the measure of the empty set is required to be 0.
Countable additivity of a measure μ: The measure of a countable disjoint union is the same as the sum of all measures of each subset.

In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events.

Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge.