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	Riemann sum - Wikipedia, the free encyclopedia
		In mathematics, a Riemann sum is a method for approximating the values of integrals.
		The left-hand Riemann sum will be an overestimation if f is monotonically decreasing on this interval, and an underestimation if it is monotonically increasing.
		The right-hand Riemann sum will be an overestimation if the function f is monotonically increasing, and an underestimation if it is monotonically decreasing.
	en.wikipedia.org /wiki/Riemann_sum (562 words)

	Riemann integral - Wikipedia, the free encyclopedia
		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.
		In more formal language, the set of all left-hand Riemann sums and the set of all right-hand Riemann sums is cofinal in the set of all tagged partitions.
		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.
	en.wikipedia.org /wiki/Riemann_integral (2271 words)

	RIEMANN
		The RIEMANN procedure computes the "Riemann sum" (or its inverse) which helps implement the backprojection operator used to reconstruct the cross-section of an object, given projections through the object from multiple directions.
		The inverse Riemann sum, which evaluates the projections given a slice through an object, is also a discrete approximation to the Radon transform.
		The angle of the ray sums from the vertical.
	www.rrz.uni-hamburg.de /RRZ/Software/IDL/HTML-Dokumentation/idl181.htm (733 words)

	7.1. Riemann Integral
		The lower sum is increasing with respect to refinements of partitions, i.e.
		The upper sum is decreasing with respect to refinements of partitions, i.e.
		In other words, the lower sum is always less than or equal to the upper sum, and the upper sum is decreasing with respect to a refinement of the partition while the lower sum is increasing with respect to a refinement of the partition.
	web01.shu.edu /projects/reals/integ/riemann.html (1802 words)

	Riemann Sums
		Riemann sums can be used to approximate the value of an area (definite integral) by filling the area, as well as possible, with rectangles, and then adding the areas of all of the rectangles.
		Although much better approximations exist (see the Trapezoidal Rule and Simpson's Rule), the Riemann sum is easy to understand and is fundamental to the concept of the definite integral.
		Riemann sums can be arduous, tedious, repetitive, and error-prone to calculate by hand, so they are prime candidates for computer software.
	www.batesville.k12.in.us /physics/CalcNet/Calculator/ti_89/riemann.htm (435 words)

	Visual Calculus - Areas - 2 (Site not responding. Last check: 2007-11-07)
		Then this is followed by showing how by increasing the number of equal-sized intervals the sum of the areas of circumscribed rectangles can better approximate the area A.
		The following table indicates the sums of the various areas together with their averages.
		It appears that the averages of the two sums may have 2 as "its limit".
	archives.math.utk.edu /visual.calculus/4/areas.2 (109 words)

	[No title]
		Since the function is increasing and one is taking the right hand endpoint for the height of the rectangular approximations, we are generating an upper sum which should converge downward to the value of the area.
		In general, one can only say that all of the Riemann sums for a particular partition are boxed in by the upper sums and the lower sums for that particular partition.
		But Riemann says that when the function y = f(x) is continuous, then the values of all of the Riemann sums approach a single limiting value as the mesh size decreases to zero.
	orion.math.iastate.edu /hentzel/class.166.ICN/Jan.17 (843 words)

	Riemann Sums, Midpoint Rule and Trapezoidal
		Use the left Riemann sum with n = 25, 50, and 100 to approximate the value of the integral.
		Use the Lower Riemann Sum to approximate the value of the integral.
		Use the Upper Riemann Sum to approximate the value of the integral.
	math.fullerton.edu /mathews/n2003/RiemannSumMod.html (325 words)

	Symbolic and Graphical Investigation of Riemann Sums (Site not responding. Last check: 2007-11-07)
		An approximation for the definite integral is determined by the sum of the signed areas of the n rectangles.
		A computer algebra system cannot be used to compute a Riemann sum in closed form for an arbitrary partition of the interval, however, if we use a consistent method for choosing the points in the subintervals having equal widths, we can obtain a closed form expression for the Riemann sum.
		The left and right sums approximate the integral with an error that is proportional to 1/n and each error is approximately the same, but of opposite sign.
	mathdemos.gcsu.edu /rsums_symbolic/rsums_symbolic_main.html (1944 words)

	The Riemann Integral
		The upper and lower sums where the interval [-1, 1] is subdivided into 10 equally spaced subintervals.
		Find conditions for a function so that the upper sum can be computed by always taking the left endpoint of each subinterval of the partition, or conditions for always being able to take the right endpoints.
		Note that upper and lower sums depend on the particular partition chosen, while the upper and lower integrals are independent of partitions.
	pirate.shu.edu /projects/reals/integ/riemann.html (1676 words)

	trapezoid.html
		Since the integral is defined to be a limit of Riemann sums, for n sufficiently large the value of the Riemann sum should be reasonably close to that of the integral.
		If you look at the picture of the Riemann approximation, you will see that the essence of the Right-Hand Riemann Sum Rule is that we approximate y=f(x), on each subinterval, by a horizontal line--the graph of a constant function.
		The function is the same one you used in your project on Riemann sums.
	www.math.uga.edu /calclab/calclab2210/trapezoid1.html (1627 words)

	Summary: The Integral
		The integral of the sum of two functions is the sum of the individual integrals, and the integral of the difference of two functions is the difference of their integrals.
		Because the derivative of a sum is the sum of the derivatives, and similarly for differences and constant multiples.
		To approximate the definite integral, we use a Riemann sum with a large number of subdivisions.
	www.zweigmedia.com /ThirdEdSite/Calcsumm6.html (901 words)

	Area Between Two Curves (Site not responding. Last check: 2007-11-07)
		Objective: The basis for determination of area under a curve and areas between two curves is the successive approximation of the area using Riemann sums over an appropriate partition of an interval.
		The actual area is determined by taking the limit of Riemann sums as the number of rectangles increases without bound in such a way that the norm of the partition approaches zero.
		The purpose of this demo is to graphically illustrate Riemann sum approximations of areas between two curves and the limiting behavior of the approximation.
	mathdemos.gcsu.edu /rsums (949 words)

	riemann.html
		Now that we have defined the right Riemann sum as a function of n, the number of subintervals, we can easily compute the right Riemann sum for various values of n, to get an idea of the limit of the right Riemann sum as n approaches infinity.
		Define left and right-hand Riemann sum approximations of the area between this curve and the x-axis, from x=1 to x=2, using n rectangles of equal width.
		The third column will contain the right Riemann sums, the fourth column will contain the error between the left Riemann sum and the integral, and the fifth column will contain the error between the right Riemann sum and the integral.
	www.math.uga.edu /calclab2210/riemann1.html (2382 words)

	PHSchool - AP* Lesson Plans
		The emphasis is on the interplay between the geometric and analytic information and on the use of calculus both to predict and to explain the observed local and global behavior of a function.
		Use of Riemann sums and the Trapezoidal Rule to approximate definite integrals of functions represented algebraically, geometrically, and by tables of values.
		In that case, the value of the definite integral is the exact area under the curve, and a Riemann sum is a rectangular approximation of that area.
	www.phschool.com /atschool/calculus/ap_calc_resources/lesson_plan_week16.html (1016 words)

	Riemann Sums and Nets (Site not responding. Last check: 2007-11-07)
		The Riemann sum is f(0)×(1-0) = 0, a rather poor estimate.
		When computing the Riemann sum, include only those points that are less than one unit away from the origin, i.e.
		When computing a Riemann sum, the function f() can be a complex or vector function.
	www.mathreference.com /ca-int,sum.html (365 words)

	Riemann Sums Using Excel and Graphing Calculators
		Column B magically fills with the values of our function at the points named in column A. The left Riemann sum is equal to 0.1 times the sum of the entries in cells B1 through B20.
		The right Riemann sum is equal to 0.1 times the sum of the entries in cells B2 through B21.
		The left Riemann sum is 0.171305 and the right Riemann sum is 0.148091.
	www.mtholyoke.edu /courses/quenell/f2001/ma101/demos/riemann.html (828 words)

	Programs for teaching calculus with TI calculators
		It is useful for teaching the concept of the definite integral and for motivating the use of the integral to solve problems that are approximated by taking sums.
		It is useful for useful for sections looking at numeric integration, particularly when dealing with convergence of the sum and the size of the error.
		The program asks for a function, an upper and lower bound for x, an initial number of subintervals, and the number of times the number of subinterval is to be doubled.
	euler.slu.edu /courseware/TIProgs/TIProgs.html (445 words)

	[No title]
		Previous Assignment p377: (6),(16),(20) Page 377 Problem 6 You are given a function f, an interval, partition points, and a description of the points xi* within the ith subinterval.
		RIEMANN SUM ============ The definition of a Riemann Sum for a function y = f[x] on an interval [a,b].
		In class Problems: (a) Show that for the constant function f(x) = c, the value of the Riemann sum is c(b-a) for any partition and any choice of x*.
	orion.math.iastate.edu /hentzel/class.166.ICN/Jan.22 (487 words)

	5 (Site not responding. Last check: 2007-11-07)
		If the limit of the Riemann sum exists, it will have the same value if f is evaluated at any point in the ith subinterval.
		Riemann sum and the process of finding the limit of this sum is called
		f is continuous, the limit of the Riemann sum above will exist as the same finite number no matter how the sample points or the intervals are chosen as long as the width of the largest subinterval goes to zero.
	www.gpc.edu /~jcraig/calc2_ch5/5s2_definite_integral.htm (350 words)

	Calc 1: Riemann sums on TI-83
		The goal here is to have TI-83 compute the various Riemann sums (left, right, midpoint) that approach the definite integral of a given function f(t) on the interval [a,b].
		For each sum, one needs to specify the number of subintervals, n, and the type of the sum -- left, right, or midpoint -- depending on where the function is to be evaluated on each subinterval of length h=(b-a)/n.
		The idea is simple: generate the list of the function values for the appropriate sum, compute its sum, and multiply it by h=(b-a)/n.
	www.fau.edu /~ebelogay/teach/2311/ti83riem.html (705 words)

	CSC3.html (Site not responding. Last check: 2007-11-07)
		To find the error, we have to compare the area under the graph with the sum of the areas of the rectangles.
		The Riemann sum with 11 rectangles of equal width underestimates the integral with an absolute error of
		with 55 rectangles of equal width and the Riemann sum using the 55 rectangles of the data, comes from the difference in the width of the rectanglesÑequal in the first case, varying in the second.
	www.dartmouth.edu /~math3f98/csc_archive/CSC3part1/CSC35.html (305 words)

	Riemann Sums - HMC Calculus Tutorial (Site not responding. Last check: 2007-11-07)
		The upper and lower sums may be considered specific Riemann sums.
		As we decrease the widths of the rectangles, we expect to be able to approximate the area of R better.
		You can create a partition of the interval and view an upper sum, a lower sum, or another Riemann sum using that partition.
	www.math.hmc.edu /calculus/tutorials/riemann_sums (456 words)

	Riemann Sums (Site not responding. Last check: 2007-11-07)
		Explicit formulas for the Riemann sums suggest algebraic combinations that lead to the trapezoid and Simpson's rules as a way to improve the rate of convergence.
		One of the main stumbling blocks with using Riemann sums to define the definite integral for beginning calculus students has been the difficulty in expressing the Riemann sum in a closed form that permits the evaluation of the limit.
		Computer algebra systems such as Derive, Mathematica, and Maple, however, can easily simplify Riemann sums for all polynomials and for functions of the form sin(mx), cos(mx), and exp(mx), thereby providing a wealth of examples involving the standard functions of calculus that students can use to investigate the convergence properties of these sums.
	ecademy.agnesscott.edu /~lriddle/riemann/riemann.htm (376 words)

	Visual Calculus - Riemann Sums - 4
		Graphically illustrate the definition of Riemann Sums for the function, y = f(x) with domain [a, b], whose graph is
		The definition of Riemann Sums will be given in the development that follows.
		and the sum of the areas of the rectangles is then:
	archives.math.utk.edu /visual.calculus/4/riemann_sums.4 (211 words)

	Riemann Sums, Simpson's Rule
		The graphing screen is large and you will have to move from this text screen to the graphing screen by means of the mouse while learning how to use the applet.
		The type, the integer N, and the numerical value of the associated riemann sum are printed in the text area.
		However, the numerical values of the trapezoidal sums and the simpson sums illustrate the relative accuracy of the simpson algorithm.
	www.math.ucla.edu /~ronmiech/Java_Applets/Riemann (819 words)

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