# A report onIntegral

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

- Integral

## 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.

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.

## 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

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.

## 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.

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

## 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 fundamental theorem of calculus relates antidifferentiation with integration.

## 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).

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

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

## 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.

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.

## Cavalieri's principle

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As follows:

As follows:

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.

## 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.

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

## Fundamental theorem of calculus

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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).