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Line Integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear integral are also used; contour integral as well, although that is typically reserved for line integrals in the complex plane. The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on intervals. Many simple formulae in physics (for example, W = F · s) have natural continuous analogs in terms of line integrals (W = ∫C F · ds)
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Calculus
Calculus
Calculus
(from Latin
Latin
calculus, literally 'small pebble', used for counting and calculations, as on an abacus)[1] 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 (concerning rates of change and slopes of curves),[2] and integral calculus (concerning accumulation of quantities and the areas under and between curves).[3] These two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. Generally, modern calculus is considered to have been developed in the 17th century by Isaac Newton and Gottfried Wilhelm Leibniz
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Shell Integration
Shell integration
Shell integration
(the shell method in integral calculus) is a means of calculating the volume of a solid of revolution, when integrating along an axis parallel to the axis of revolution. This is in contrast to disk integration which integrates along the axis perpendicular to the axis of revolution.Contents1 Definition 2 Example 3 See also 4 ReferencesDefinition[edit] The shell method goes as follows: Consider a volume in three dimensions obtained by rotating a cross-section in the xy-plane around the y-axis. Suppose the cross-section is defined by the graph of the positive function f(x) on the interval [a, b]
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Faà Di Bruno's Formula
Bruno's Supermarkets, LLC was an American chain of grocery stores with its headquarters in Birmingham, Alabama.[1] It founded in 1932 by Joe Bruno in Birmingham. During the company's pinnacle, it operated over 300 stores under the names Bruno’s, Food World, Foodmax, Food Fair, Fresh Value, Vincent's Markets, Piggly Wiggly, Consumer Foods, and American Fare in Alabama, Florida, Georgia, Mississippi, Tennessee, and South Carolina. The chain was acquired by Birmingham-based Belle Foods
Belle Foods
which discontinued the brand in 2012.Contents1 History 2 Bankruptcy 3 Bruno's as a defunct brand 4 Sports Sponsorship 5 References 6 External linksHistory[edit] The company began as a market opened by Joseph Bruno in Birmingham, Alabama
Alabama
during the Great Depression
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Integral
In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of calculus, with its inverse, differentiation, being the other. Given a function f of a real variable x and an interval [a, b] of the real line, the definite integral ∫ a b f ( x ) d x displaystyle int _ a ^ b !f(x),dx is defined informally as the signed area of the region in the xy-plane that is bounded by the graph of f, the x-axis and the vertical lines x = a and x = b. The area above the x-axis adds to the total and that below the x-axis subtracts from the total. Roughly speaking, the operation of integration is the reverse of differentiation
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Lists Of Integrals
Integration is the basic operation in integral calculus. While differentiation has easy rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful
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Antiderivative
In calculus, an antiderivative, primitive function, primitive integral or indefinite integral[Note 1] of a function f is a differentiable function F whose derivative is equal to the original function f
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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, ∞ displaystyle infty , − ∞ displaystyle -infty , or in some instances as both endpoints approach limits
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Riemann Integral
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göttingen
University of Göttingen
in 1854, but not published in a journal until 1868.[1] For many functions and practical applications, the Riemann integral
Riemann integral
can be evaluated by the fundamental theorem of calculus or approximated by numerical integration. The Riemann integral
Riemann integral
is unsuitable for many theoretical purposes
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Lebesgue Integration
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the x-axis. The Lebesgue integral extends the integral to a larger class of functions. It also extends the domains on which these functions can be defined. Long before the advent of the 20th century, mathematicians already understood that for non-negative functions with a smooth enough graph—such as continuous functions on closed bounded intervals—the area under the curve could be defined as the integral, and computed using approximation techniques on the region by polygons
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Methods Of Contour Integration
In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.[1][2][3] Contour integration
Contour integration
is closely related to the calculus of residues,[4] a method of complex analysis. One use for contour integrals is the evaluation of integrals along the real line that are not readily found by using only real variable methods.[5]
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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 their derivative and antiderivative. It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found
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Disc Integration
Disc integration, also known in integral calculus as the disc method, is a means of calculating the volume of a solid of revolution of a solid-state material when integrating along an axis "parallel" to the axis of revolution. This method models the resulting three-dimensional shape as a stack of an infinite number of discs of varying radius and infinitesimal thickness. It is also possible to use the same principles with rings instead of discs (the "washer method") to obtain hollow solids of revolutions
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Integration By Substitution
In calculus, integration by substitution, also known as u-substitution, is a method for finding integrals. Using the fundamental theorem of calculus often requires finding an antiderivative. For this and other reasons, integration by substitution is an important tool in mathematics. It is the counterpart to the chain rule of differentiation.Contents1 Substitution for single variable1.1 Proposition 1.2 Proof 1.3 Examples1.3.1 Example 1: from right to left 1.3.2 Example 2: from left to right 1.3.3 Example 3: antiderivatives2 Substitution for multiple variables 3 Application in probability 4 See also 5 References 6 External linksSubstitution for single variable[edit] Proposition[edit] Let I ⊆ ℝ be an interval and φ : [a,b] → I be a differentiable function with integrable derivative. Suppose that ƒ : I → ℝ is a continuous function
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Inverse Functions And Differentiation
In mathematics, the inverse of a function y = f ( x ) displaystyle y=f(x) is a function that, in some fashion, "undoes" the effect of f displaystyle f (see inverse function for a formal and detailed definition). The inverse of f displaystyle f is denoted f − 1 displaystyle f^ -1 . The statements y = f(x) and x = f −1(y) are equivalent. Their two derivatives, assuming they exist, are reciprocal, as the Leibniz notation
Leibniz notation
suggests; that is: d x d y ⋅ d y d x = 1. displaystyle frac dx dy ,cdot , frac dy dx =1
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Trigonometric Substitution
In mathematics, Trigonometric substitution
Trigonometric substitution
is the substitution of trigonometric functions for other expressions. One may use the trigonometric identities to simplify certain integrals containing radical expressions:[1][2]Substitution 1. If the integrand contains a2 − x2, let x = a sin ⁡ θ displaystyle x=asin theta and use the identity 1 − sin 2 ⁡ θ = cos 2 ⁡ θ . displaystyle 1-sin ^ 2 theta =cos ^ 2 theta . Substitution 2
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