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Topic: Riemann integral

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	Riemann integral - Wikipedia, the free encyclopedia
		While the Riemann integral is unsuitable for many theoretical purposes, it is one of the easiest integrals to define.
		The definition of the Lebesgue integral is not obviously a generalization of the Riemann integral, but it is not hard to prove that every Riemann-integrable function is Lebesgue-integrable and that the values of the two integrals agree whenever they are both defined.
		An integral which is in fact a direct generalization of the Riemann integral is the Henstock-Kurzweil integral.
	en.wikipedia.org /wiki/Riemann_integral (2265 words)

	Integral (Site not responding. Last check: 2007-11-05)
		In calculus, the integral, of a function, is the size of the area bounded by the x-axis and the graph of a function, f(x); negative areas are possible.
		The concept of Riemann integration was developed first, and Lebesgue integrals were developed to deal with pathological cases for which the Riemann integral was not defined.
		Both the Riemann and the Lebesgue integral are approaches to integration which seek to measure the area under the curve, and the overall schema in both cases is the same.
	bopedia.com /en/wikipedia/i/in/integral.html (761 words)

	Reference.com/Encyclopedia/Integral
		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.
		The Riemann integral was created by Bernhard Riemann in 1854 and was the first rigorous definition of the integral.
		The Riemann-Stieltjes integral, an extension of the Riemann integral.
	www.reference.com /browse/wiki/Integral (1416 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.
		The only reason that the Dirichlet function is Lebesgue, but not Riemann, integrable, is that its spikes occur on the rationals, a set of numbers which is, in comparison to the irrational numbers, a very small set.
	planetmath.org /encyclopedia/ComparisonBetweenLebesgueAndRiemannIntegration2.html (1078 words)

	Riemann integral at opensource encyclopedia (Site not responding. Last check: 2007-11-05)
		In a branch of mathematics known as real analysis, the Riemann integral is a simple way of viewing the integral of a function on an interval as the area under the curve.
		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 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.wiki.tatet.com /Riemann_integral.html (1146 words)

	ipedia.com: Integral Article (Site not responding. Last check: 2007-11-05)
		The integral value of a real number x is defined as the largest integer which is less than, or equal to, x.
		Improper integrals usually turn up when the range of the function is infinite or, in the case of the Riemann integral, when the domain is infinite.
		The Riemann integral was created by Bernhard Riemann and was the first rigorous definition of the integral.
	www.ipedia.com /integral.html (1470 words)

	Math Tutor Level 3 Text choice1=Integral choice2=Theory choice3=Introduction
		Therefore, for negative functions, the Riemann integral is equal to minus the geometric area of the region between the graph of f and the x-axis.
		For a general function, the Riemann integral is equal to the mathematical area of the region between the graph of f and the x-axis, that is, the geometric area of the parts above the x-axis minus the geometric area of the parts below the x-axis.
		The fact that Riemann integrability is not hurt by a finite number of discontinuities is related to the fact that the value of Riemann integral is not influenced by a change of the integrated function at a finite number of points.
	math.feld.cvut.cz /mt/txtd/1/txe3da1a.htm (1801 words)

	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.
		With this definition, an integral can exist when f and g share points of discontinuity, as long as they are not discontinuous from the same side at the same point.
		However, g may have jump discontinuities, or may have derivative zero almost everywhere while still being continuous and increasing (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 (545 words)

	Integral Article, Integral Information (Site not responding. Last check: 2007-11-05)
		Intuitively, the integral of a continuous, positive real-valued function f of one real variable x between aleft endpoint a and a right endpoint b represents the area bounded by the lines x=a, x=b, thex-axis, and the curve defined by the graph of f.
		There are also many less commonways of calculating definite integrals; for instance, Parseval's identity can be used to transform the integral of a square into an infinite sum.Occasionally an integral can be evaluated by a trick; for an example of this, see Gaussian integral.
		An integral which can only be evaluated by considering it asthe limit of integrals on successively larger and larger integrals is called an improper integral.
	www.anoca.org /integrals/integration/integral.html (1330 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		Contents of the course: Although the classical Riemann integral is extremely useful for explicit calculations (due to its close connection with differentiation), it suffers from a number of technical disadvantages, e.g.
		The main difference between Riemann and Lebesgue integration can be described as follows: for the Riemann integral the domain of the function is decomposed into small intervals, on which the function is almost constant, and the integral is approximated by the sum of the areas of the resulting `rectangles'.
		For the Lebesgue integral the range of the function is divided into small intervals, and the integral is again approximated by sums of areas of rectangles, but the bases of these rectangles are now much more complicated sets than before (in particular they are no longer intervals).
	www.mat.univie.ac.at /~kschmidt/SS04e.html (301 words)

	riemann integral (Site not responding. Last check: 2007-11-05)
		For example, it was recognized that the Riemann integral has poor convergence properties; for example, a function which is the pointwise limit of a uniformly bounded sequence of integrable functions need not be Riemann integrable.
		There are two distinct integrals which are defined with a slight adjustment in the definition of the Riemann integral.
		The McShane integral is equivalent to the Lebesgue integral, and overcomes the convergence problems of the Riemann integral.
	fym.la.asu.edu /~pvaz/seminars/riemann.html (178 words)

	Riemann (Site not responding. Last check: 2007-11-05)
		Bernhard Riemann (1826-1866) studied at Gottingen and Berlin, earning his Ph.D. in 1851 at Gottingen for a thesis dealing with complex function theory and what are now called Riemann Surfaces.
		Leibniz defined an integral as a sum and his notation reveals his understanding of the integral as a sum (his integral sign is just a long S) of infinitesimally small rectangles of height y and width dx.
		Riemann is most famous for the Riemann Hypothesis, the most famous unsolved problem in mathematics today.
	www.dean.usma.edu /math/people/rickey/class/Riemann.html (529 words)

	Rieman Summs
		An integral computes the area under some arbitrary curve, given by a function.When a shape is complex, like our example of the Salton Sea, we can approximate the area by breaking up the region into smaller pieces whose areas are easily calculated, such as squares or rectangles.
		However, when an integral is defined over a specific interval, as stated above for the Riemann integral, then there are a number of methods for finding approximate solutions to the integral.
		The Riemann integral defined above was shown to represent the area under a function on a specified interval.
	www-rohan.sdsu.edu /~jmahaffy/courses/f00/math122/lectures/riemann_sums/riemanns.html (2113 words)

	7.1. Riemann Integral (Site not responding. Last check: 2007-11-05)
		Note that upper and lower sums depend on the particular partition chosen, while the upper and lower integrals are independent of partitions.
		The third example shows that not every function is Riemann integrable, and the second one shows that we need an easier condition to determine integrability of a given function.
		Suppose f is Riemann integrable over an interval [-a, a] and f is an odd function, i.e.
	web01.shu.edu /projects/reals/integ/riemann.html (1802 words)

	The Riemann Integral
		Riemann sums have the practical disadvantage that we do not know which point to take inside each subinterval.
		Is the Dirichlet function Riemann integrable on the interval [0, 1] ?
		Suppose f is a Riemann integrable function defined on [a, b].
	pirate.shu.edu /projects/reals/integ/riemann.html (1676 words)

	No Title
		Let's trace this development of the integral as a rough and ready way to solve problems of physics to a full-fledged theory.
		And this issue is to become central to the concept of integral.
		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.
	www.math.tamu.edu /~don.allen/history/riemann/riemann.html (1521 words)

	Inequalities for Riemann-Stieltjes Integrals
		J.V. Herod, A Gronwall inequality for linear Stieltjes integrals, Proc.
		Darst and H. Pollard, An inequality for the Riemann-Stieltjes integral, Proc.
		Groh, A nonlinear Volterra-Stieltjes integral equation and a Gronwall inequality in one dimension, Illinois J. Math., 24(2) (1980), 244-263.
	rgmia.vu.edu.au /ineq_riemann.htm (524 words)

	Integral
		The integral value of a real number x is defined to be the largest integer which is less than or equal to x; it is often denoted by ⌊x⌋ and also called the floor function.
		In the integral calculus, the integral of a function is informally defined as the size of the area delimited by the x axis and the graph of the function.
		Details can be found under Riemann integral and Lebesgue integral.
	www.wordlookup.net /in/integral.html (883 words)

	PlanetMath: Riemann integral
		be the infimum of the set of upper Riemann sums with each partition in
		be the supremum of the set of lower Riemann sums with each partition in
		This is version 6 of Riemann integral, born on 2001-10-19, modified 2005-02-26.
	planetmath.org /encyclopedia/RiemannIntegral.html (82 words)

	An Introduction to the Gauge Integral
		The Riemann integral is simpler to define than any of the other integrals discussed below, and it is the "standard" integral that we teach to undergraduate students.
		Neither the improper Riemann integral nor the Lebesgue integral yielded a fully satisfactory construction of antiderivatives.
		It was already hard enough with the Riemann integral -- for that integral we had to use rather bizarre functions, such as the characteristic function of the rationals.
	www.math.vanderbilt.edu /~schectex/ccc/gauge (4371 words)

	Class diary for Math 311:01, spring 2003
		II) Suppose that f is defined on the interval [a,c], and that f is Riemann integrable on [a,b] and that f is Riemann integrable on [b,c].
		Definition The Upper Riemann integral is the inf of all of the upper sums of f.
		The Lower Riemann integral is the sup of all of the lower sums of f.
	www.math.rutgers.edu /~greenfie/currentcourses/math311/diary.html (9981 words)

	Lebesgue integral -- Britannica Concise Encyclopedia - The online encyclopedia you can trust! (Site not responding. Last check: 2007-11-05)
		Lebesgue sums are used to define the Lebesgue integral of a bounded function by partitioning the y-values instead of the x-values as is done with Riemann sums.
		The Lebesgue integral is the concept of the measure (q.v.
		His developments in geometry were precursors to integral calculus, especially his method of indivisibles, a means of determining the size of geometric figures.
	www.britannica.com /ebc/article-9047549 (986 words)

	[No title]
		Cauchy's integral was successful in its goal of integrating continuous functions, but the budding theory of Fourier series prompted the need to integrate a much larger class of functions.
		Riemann's modification of Cauchy's approach (which is the integral we teach in introductory calculus) represented a major step forward.
		The Riemann integral was capable of handling functions with many discontinuities although this set still needed to be relatively small.
	web.centre.edu /mat/kymaa/meetings/abstracts03.doc (1628 words)

	An Open Letter to Authors of Calculus Books
		The gauge integral (also known as the generalized Riemann integral, the Henstock integral, the Kurzweil integral, the Riemann complete integral, etc.) was discovered later, but it is a "better" integral in nearly all respects.
		The idea of introducing the gauge integral to college freshmen is not entirely new; it was promoted, for instance, in the paper "The Teaching of the Integral" by Bullen and Výborný, Journal of Mathematical Education in Science and Technology, vol.
		A book covering the gauge integral might still include the characteristic function of the rationals, as an example of a bounded function that is gauge integrable but not Riemann integrable on a closed bounded interval.
	www.math.vanderbilt.edu /~schectex/ccc/gauge/letter (2241 words)

	Math Tutor Level 3 Text choice1=Integral choice2=Theory choice3=Introduction
		To start with, the Riemann integral is a definite integral, therefore it yields a number, whereas the Newton integral yields a set of functions (antiderivatives).
		The Riemann integral is a geometric notion (area), while the Newton integral is an algebraic notion.
		We see that a continuous function is both Riemann integrable and Newton integrable, and we can get an antiderivative (the Newton integral) using the Riemann integral.
	math.feld.cvut.cz /mt/txtd/1/txe3da1d.htm (631 words)

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