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	Riemann-Stieltjes integral - Wikipedia, the free encyclopedia
		In mathematics, the Riemann-Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes.
		For another formulation of integration that is much more general, see Lebesgue integration.
		However, g may have jump discontinuities, or may have derivative zero almost everywhere while still being continuous and nonconstant (for example, g could be the Cantor function or the question mark function), in either of which cases the Riemann-Stieltjes integral is not captured by any expression involving derivatives of g.
	en.wikipedia.org /wiki/Riemann-Stieltjes_integral (512 words)

	Lebesgue integration - Wikipedia, the free encyclopedia
		The Riemann integral, proposed by Bernhard Riemann (1826-1866), is a broadly successful attempt to provide such a foundation for the integral.
		Riemann's definition starts with the construction of a sequence of easily-calculated integrals which converge to the integral of a given function.
		The upper Riemann sum is the sum of the upper volumes of all the slices, and similarly for the lower Riemann sum.
	en.wikipedia.org /wiki/Lebesgue_integration (2678 words)

	Henri Lebesgue (Site not responding. Last check: 2007-10-29)
		Integration is a mathematical operation that corresponds to the informal idea of finding the area under the graph of a function.
		The first theory of integration was developed by Archimedes in the third century BC with his method of quadratures, but this could be applied only in limited circumstances with a high degree of geometric symmetry.
		The Riemann integral had been generalised to the improper Riemann integral to measure functions whose domain of definition was not a closed interval.
	www.free-download-soft.com /info/adult-add-adult.html (689 words)

	PlanetMath: Comparison between Lebesgue and Riemann Integration
		The Riemann and Lebesgue integral are defined in different ways, with the latter generally perceived as the more general.
		Clearly, the upper and lower Riemann sums converge to 1 and 0, respectively, in the limit of the size of largest interval in the partition going to zero.
		This is version 1 of Comparison between Lebesgue and Riemann Integration, born on 2005-08-24.
	planetmath.org /encyclopedia/ComparisonBetweenLebesgueAndRiemannIntegration2.html (1078 words)

	Riemann integral (Site not responding. Last check: 2007-10-29)
		An improvement is to use the Lebesgue integral which both succeeds at integrating a broader variety of functions, as well as better describing the interactions of limits and integrals.
		The fact that the seemingly disparate theories of Riemann integration and antiderivatives are essentially talking about the same subject is contained in the fundamental theorem of calculus.
		The Riemann integral is defined to be the only number that is less than or equal to all upper sums (as the partition varies); it is both the infimum of the set of all upper sums and the supremum of the set of all lower sums.
	www.sciencedaily.com /encyclopedia/riemann_integral (1295 words)

	Lebesgue integration - Wikipedia
		In the mathematical branch of analysis, Lebesgue integration is a means of extending the usual notion of (Riemann-)integration to more functions and to more general settings.
		Integration is the mathematical operation which corresponds to finding the area under a function.
		Furthermore, the extension of the notion of integration to functions defined on general measure spaces, which include probability spaces, allows the proper formulation of the foundations of probability and statistics.
	nostalgia.wikipedia.org /wiki/Lebesgue_integration (832 words)

	Riemann integration (Site not responding. Last check: 2007-10-29)
		An improvement is to use the Lebesgue integral which both succeeds at integrating a broader variety of functions, as well as betterdescribing the interactions of limits and integrals.
		The fact that the seemingly disparate theories of Riemann integration and antiderivatives are essentially talking about thesame subject is contained in the fundamental theorem of calculus.
		The Riemann integral is defined to be the only number that is less than or equal to all upper sums (as the partition varies);it is both the infimum of the set of all upper sums and the supremum of the set of all lower sums.
	www.therfcc.org /riemann-integration-32820.html (1198 words)

	Riemann integral - Wikipedia, the free encyclopedia
		In a branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann (pronounced REE mahn), was the first rigorous definition of the integral of a function on an interval.
		So we will make an alternate definition of the Riemann integral which is easier to work with, then prove that it is the same as the definition we have just made.
		If we use regular subdivisions and left-hand or right-hand Riemann sums, then the two terms on the left are equal to zero, since every endpoint except 0 and 1 will be irrational, but as we have seen the term on the right will equal 1.
	www.wikipedia.org /wiki/Riemann_integral (2255 words)

	Integral Computing integrals External links sum long s Integration by parts explicit formulae range Riemann integral ... (Site not responding. Last check: 2007-10-29)
		The ∫ sign represents integration, a and b are the endpoints of the interval, f(x) is the function we are integrating, and dx is a notation for the variable of integration.
		If the domain of the function to be integrated is the real numbers, and if the region of integration is an interval, then the greatest lower bound of the interval is called the lower limit of integration, and the least upper bound is called the upper limit of integration.
		Improper integrals usually turn up when the range of the function to be integrated is infinite or, in the case of the Riemann integral, when the domain of the function is infinite.
	en.powerwissen.com /%7C%7CSL%7C%7C%7C%7CSL%7C%7CJ7ac85hTtIT3LgTSBcWA%3D%3D_Integral.html (1525 words)

	Lebesgue integration -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-29)
		Lebesgue integration is a mathematical (A well-substantiated explanation of some aspect of the natural world; an organized system of accepted knowledge that applies in a variety of circumstances to explain a specific set of phenomena) theory that defines the integral for a very large class of functions.
		It had long been understood that for functions with a smooth enough graph (such as continuous functions on closed bounded intervals) the area under the curve could be defined and computed using techniques of approximation of the region by polygons.
		The (Click link for more info and facts about Riemann integral) Riemann integral, proposed by (Pioneer of non-Euclidean geometry (1826-1866)) Bernhard Riemann (1826-1866), is a broadly successful attempt to provide such a foundation for the integral.
	www.absoluteastronomy.com /encyclopedia/l/le/lebesgue_integration.htm (3007 words)

	Yes, But Will it Fly? by Martin Davis and Matt Insall in the Nexus Network Journal vol. 4 no. 4 (Autumn 2002)
		Hamming about the physical effect of Lebesgue and Riemann integrals and whether it made a difference whether one or the other was used, for example, in the design of an airplane.
		Similarly, when computing that particular integral using Lebesgue integration, if the value obtained is different from that obtained using Riemann integration, then the values elsewhere in the design will be skewed.
		integration may have less practice with Riemann integration, having spent less time doing these integrals quickly than they have spent learning the theory of Lebesgue integration over the last few years of their training.
	www.zblmath.fiz-karlsruhe.de /exx/journals/NNJ/DavIns-HM.html (1421 words)

	Riemann-Stieltjes integral (Site not responding. Last check: 2007-10-29)
		In mathematics, the Riemann-Stieltjes integral is a generalization of the Riemann integral.
		For another formulation of integration that is more general, see Lebesgue integration.
		However, g may have jump discontinuities, or may have derivative zero almost everywhere while still being continuous and nonconstant (for example, g could be the celebrated Cantor function), in either of which cases the Riemann-Stieltjes integral is not captured by any expression involving derivatives of g.
	www.sciencedaily.com /encyclopedia/riemann_stieltjes_integral (418 words)

	Background
		This is analogous to integrating the differential equation, and is as opposed to something like a spectral or finite-difference method, where the differential equation is approximated by a set of simultaneous linear equations in points all throughout the domain.
		Integration methods are most appropriate for my particular problem for a number of reasons.
		I mentioned before that trapezoidal integration is a special case of second-order RK (one can see now that "second order" refers to the power of dt that is in agreement with the actual function).
	archive.ncsa.uiuc.edu /Classes/MATH198/rgiordan/numerics.html (759 words)

	Real analysis/Riemann integration - Wikibooks
		Riemann integration is the formulation of integration most people think of if they ever think about integration.
		It is the only type of integration considered in most calculus classes; many other forms of integration, notably Lebesgue integrals, are extensions of Reimann integrals to larger classes of functions.
		The Riemann integral was developed by Bernhard Riemann in 1854 and was the first rigorous definition of integration.
	en.wikibooks.org /wiki/Real_analysis/Riemann_integration (153 words)

	No Title
		The validity of term-by-term integration was lacking until until Cauchy proved conditions for it to hold.
		Bernhard Riemann (1826-66) no doubt acquired his interest in problems connected with trigonometric series through contact with Dirichlet when he spent a year in Berlin.
		For his Habilitationsschrift (1854) Riemann under-took to study the representation of functions by trigonometric functions.
	www.math.tamu.edu /~don.allen/history/riemann/riemann.html (1521 words)

	[No title]
		Whereas Riemann's procedure is still ruled by the intuition of continuity, Lebesgue makes integration of discontinuous functions "natural"; the convergence theorem assures the superiority of his methods (IV).
		The consideration of invariance properties for integration by Riemann and Poincare leads to the birth of analysis situs (VIII).
		For Michel, the topology of integration theory is therefore mainly the topology of compactness (XI).
	www.math.niu.edu /~rusin/known-math/99/hist_integ (2511 words)

	Learn more about Real analysis in the online encyclopedia. (Site not responding. Last check: 2007-10-29)
		It can be seen as a rigorous version of calculus and studies concepts such as sequences and their limits, continuity, differentiation, integration and sequences of functions.
		The presentation of real analysis in advanced texts usually starts with simple proofs in elementary set theory, a clean definition of the concept of function, and an introduction to the natural numbers and the important proof technique of mathematical induction.
		Then one can do integration (Riemann and Lebesgue) and prove the Fundamental Theorem of Calculus, typically using the mean value theorem.
	www.onlineencyclopedia.org /r/re/real_analysis.html (489 words)

	wikien.info: Main_Page (Site not responding. Last check: 2007-10-29)
		Lebesgue integration is a mathematical theory that extends the integral to a very large class of functions.
		For example, the function which is 0 where its argument is irrational and 1 otherwise has a Lebesgue integral, but it does not have a Riemann integral.
		Indeed, the element of calculation for the Riemann integral is the rectangle [a, b] × [c, d], whose area is calculated to be (b-a)(c-d).
	pardus.info /index.php?title=Lebesgue_integration (2901 words)

	Noncommutative Riemann integration and singular traces for C*-algebras - Guido, Isola (ResearchIndex)
		as a suitable closure, `a la Dedekind, of A, in analogy with one of the classical characterizations of Riemann measurable functions [26], and show that A R is a C -algebra, and extends to a semicontinuous semifinite trace on A R.
		Noncommutative Riemann integration and singular traces for C - algebras.
		1 Integration theorems for gages and duality for unimodular gr..
	citeseer.ist.psu.edu /425412.html (696 words)

	Lebesgue-Stieltjes integration - Encyclopedia, History, Geography and Biography
		If you are having difficulty understanding this article, you might want to first learn more about integrals, particularly the Lebesgue integral, and measure theory.
		In measure-theoretic analysis and related branches of mathematics, the Lebesgue-Stieltjes integration generalizes the Riemann-Stieltjes and Lebesgue integration, preserving the many advantages of the latter in a more general measure-theoretic framework.
		Lebesgue-Stieltjes integrals, named for Henri Leon Lebesgue and Thomas Joannes Stieltjes, are also known as Lebesgue-Radon integrals or just Radon integrals, after Johann Radon, to whom much of the theory of the present topic is due.
	www.arikah.net /encyclopedia/Lebesgue-Stieltjes_integral (690 words)

	Mathematical analysis
		In the middle of the century Riemann introduced his theory of integration.
		The last third of the 19th century saw the arithmetization of analysis by Weierstrass, who thought that geometric reasoning was inherently misleading, and introduced the ε-δ definition of limit.
		Around that time, the attempts to refine the theorems of Riemann integration led to the study of the "size" of the discontinuity sets of real functions.
	www.brainyencyclopedia.com /encyclopedia/m/ma/mathematical_analysis.html (481 words)

	Riemann Sums - Function Integration
		The purpose of the applet below is to demonstrate how Riemann sums approximate the value of a definite integral.
		On every subinterval, one can choose either the left or right value of the function, the lower or the larger of the two, or the value at a random point on the interval, or at its midpoint.
		The applet displays both the definite integral as the function of its upper limit and its approximation by Riemann sums.
	www.cut-the-knot.org /Curriculum/Calculus/RiemannSums.shtml (126 words)

	PlanetMath: Lebesgue integral
		The Lebesgue integral equals the Riemann integral everywhere the latter is defined; the advantage to the Lebesgue integral is that it is often well defined even when the corresponding Riemann integral is undefined.
		For example, the Riemann integral of the characteristic function of the rationals in
		Cross-references: measure, rationals, Riemann integral, function, interval, Lebesgue measure, collection, finite, characteristic function, measure space, measurable function
	planetmath.org /encyclopedia/Integral2.html (153 words)

	Analysis WebNotes: Chapter 08, Class Ch08 (Site not responding. Last check: 2007-10-29)
		The definition of integration is quite complicated and we devote some time to studying that.
		We identify a class of "integrable" functions wihich is analogous to the "differentiable" functions of Chapter 4, and show that all continuous functions are integrable.
		Evaluating the integral based on the definition alone is very hard, so we prove the Fundamental theorem of calculus; the theoretical tool which is the basis of all the integration techniques in calculus.
	www.math.unl.edu /~webnotes/classes/classCh08/classCh08.htm (130 words)

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