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Topic: Trapezium rule

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	Trapezium rule - Wikipedia, the free encyclopedia
		The trapezium rule works by approximating the region under the graph of the function f(x) by a trapezium and calculating its area.
		The trapezium rule is one of a family of formulas for numerical integration called Newton-Cotes formulas.
		Moreover, the trapezium rule tends to become extremely accurate when periodic functions are integrated over their periods, a fact best understood in connection with the Euler-Maclaurin summation formula.
	en.wikipedia.org /wiki/Trapezium_rule (224 words)

	Second Year Aerospace Engineering Aerodynamics Laboratory
		The trapezium rule can be used to find the area under graphs as if it were made up of many adjacent trapeziums.
		Trapezium rule will always give an over estimate of the true integration and the fact that there was not many data points only amplifies this overestimate.
		The use of trapezium rule to carry out the integrations may have limited the accuracy of answers due to its tendency to overestimate areas.
	website.lineone.net /~rh/personal/areolab2.htm (1508 words)

	Captain and crew update Part Three
		The trapezoidal rule is a numerical method for integrating a function between two bounds.
		This rule states that the area under a curve can be approximated by a trapezium.
		The trapezoidal rule may be stated as follows: Divide the base line into any number of equal parts and erect ordinates at the points of division.
	www.hansa-online.de /print.asp?artikelID=534 (980 words)

	Quadrature
		This is a bonus of the rule; it actually integrates exactly a higher degree polynomial than we would expect.
		Halving the interval therefore reduces the overall error by 4 for the Trapezium Rule and by 16 for Simpson's Rule.
		is assumed constant, T(h) is the composite Trapezium Rule with interval h.
	www.maths.lancs.ac.uk /~gilbert/m241/node6.html (534 words)

	The Trapezium Rule
		The trapezium rule is a way of estimating the area under a curve.
		We know that the area under a curve is given by integration, so the trapezium rule gives a method of estimating integrals.
		The trapezium rule works by splitting the area under a curve into a number of trapeziums, which we know the area of.
	www.mathsrevision.net /alevel/pages.php?page=44 (123 words)

	Numerical Integration
		Trapezium Rule: This method uses an arbitrary number of equal subdivisions.
		Simpson's Rule: This method uses any even number 2N of equal subdivisions.
		Simpson's Rule is far more accurate that the Trapezium Rule, in general, for essentially the same amount of calculation.
	www.maths.abdn.ac.uk /~igc/tch/index/engbook/node44.html (110 words)

	Essay or Coursework - Area Under a Straight Line Graph - Calculate the area under a straight line graph.
		A trapezium is a quadrilateral that has 1 pair of parallel sides.
		Let a and b be the lengths of the parallel sides of a trapezium, and let h be the distance between them (measured at right-angles to them).
		The trapezium rule is a method of calculating approximately the area under a curve.
	www.coursework.info /i/34123.html (553 words)

	Numerical Analysis (Site not responding. Last check: 2007-10-22)
		The most widely known of these is the trapezium rule, which is equivalent to performing linear interpolation between the points.
		The error in the trapezium rule depends upon the value of the derivatives at the end points, so if these derivatives vanish, or if the function is being integrated over one period, the trapezium rule gives good value for money.
		In many cases, they also require the function to be evaluated at less points than the trapezium rule to achieve a given accuracy.
	www.hpcc.ecs.soton.ac.uk /training/numan_i.html (1833 words)

	Numerical Methods Lecture Notes: integration
		However, the size of the domain would be halved, thus requiring the Trapezium Rule to be evaluated twice and the contributions summed.
		The approach is to cover the domain to be integrated with a triangle or trapezium (whichever is geometrically more appropriate) as is shown in figure 17.
		The Mid-point rule is an example of this: with just a single function evaluation it obtains the same order of accuracy as the Trapezium Rule (which requires two points).
	www.damtp.cam.ac.uk /user/fdl/people/sd103/lectures/nummeth98/integration.htm (1086 words)

	Phoenician Religion -- Pagan (Site not responding. Last check: 2007-10-22)
		When their violent dispute was brought before Zeus, it was ruled that for a third part of the year Adonis was to dwell by himself; for a third part with Aphrodite; and for a third part with Persephone.
		It appears that the primitive form of this sign was a trapezium closed by a horizontal line at the top and surmounted in the middle by a circle.
		If we do attribute the name of Saeculum Frugiferum to the god seated between two sphinxes, we must admit that we do not know under what appellation he was addressed by those who remained faithful to him up to the time of the Empire, and looked to him for their prosperity.
	www.phoenicia.org /pagan.html (13430 words)

	Project \| Questions and Answers (Site not responding. Last check: 2007-10-22)
		If we apply the BACKWARD EULER method to an IVP whose true solution is y = cos x, as we did in the report, we would obtain a phase plane plot, which...
		If we apply the TRAPEZIUM RULE method to an IVP whose true solution is y = cos x, as we did in the report, we would obtain a phase plane plot, which...
		The equations on which most numerical methods perform poorly (low accuracy), so a suitable numerical method should be chosen very carefully.
	www.doc.ic.ac.uk /~pb401/DE/qna.cgi (416 words)

	Project \| Download (Site not responding. Last check: 2007-10-22)
		This implementation of the Trapezium Rule technically should work for an ODE of ANY order.
		The idea behind the implementation is to produce a set of trapezium equations for each of the variables that represent the different order derivatives, then to simultaneously solve them to get the new set of values for each of the derivatives and for any y, then continue to get a list of y points.
		We have modified the various rules, as we wish to solve Kepler's Equation, which consists of two simultaneous second-order differential equations.
	www.doc.ic.ac.uk /~pb401/DE/dl.cgi (797 words)

	The Intrinsic Method (Site not responding. Last check: 2007-10-22)
		Returning to the possibility of solving the integral equation using the trapezoidal or trapezium rule, let us consider the case of a linear differential equation, such as our examples.
		This intrinsic method is 2nd order accurate as that is the accuracy of the trapezoidal rule for integration.
		In fact the intrinsic method is also stable for the growth equation when it is analysed as discussed earlier, so that the method is in fact stable for all 3 classes of equations.
	www.sst.ph.ic.ac.uk /people/a.mackinnon/Lectures/compphys/node13.html (194 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		function [answer, table] = romberg(tvalues,r) % romberg: carries out Romberg integration using a sequence of trapezium % rule estimates as input.
		% % Usage: [answer, table] = romberg(tvalues) % Inputs: tvalues a list of trapezium rule estimates, each one produced with % twice the number of intervals as the previous one.
		Type return to stop.'; else len = length(tvalues); table = zeros(len,len); table(:,1) = tvalues(:); for j = 1:len-1 for k = j+1:len table(k,j+1) = ((r^j)table(k,j) - table(k-1,j)) / (r^j -1); end end disp('The Romberg integration table is: ') disp(table); answer = table(len,len); pstr='Provide another trapezium* rule estimate if you wish.
	www.math.ucalgary.ca /~westbroo/amat491/mscripts/romberg.m (148 words)

	CASA - integration (Site not responding. Last check: 2007-10-22)
		Remark 3 An integral transformation induces a transformation of the numerical integration rules.
		In general all rules composed form a single rule - using equi-sized subintervals - are one order less accurate than the single rule.
		The order of the numerical integration error of the midpoint rule is larger than that of the trapezoid rule.
	www.win.tue.nl /casa/education/courses/2N330/lectures/integration/integration (1623 words)

	Quadrature Error
		One way we might evaluate the error is to use Taylor expansions with integral remainders for f.
		We illustrate for the trapezium rule using the Taylor results
		An alternative and much commoner approach is to use composite rules.
	www.maths.lancs.ac.uk /~gilbert/m243b/node12.html (202 words)

	Motivate : Final part of Philip's talk (Site not responding. Last check: 2007-10-22)
		You can find out more about the Trapezium Rule if you are not very familiar with it in the Additional Notes, and you can have a go at using it in the next set of tasks if you wish.
		Now you should spend some time trying out the Trapezium Rule, if it is unfamiliar to you, and Hardy’s 39a formula on functions you have already met, and then have a go at solving our original problem.
		To do this you will need to use the information in the table and graph towards the end of the second part of this talk, together with which ever method of numerical integration you prefer.
	www.motivate.maths.org /conferences/conf51/c51_talk3.shtml (981 words)

	ARABIA - Online Information article about ARABIA (Site not responding. Last check: 2007-10-22)
		rule under conditions which precluded the use of even the simplest See also:
		By training and temperament he was better qualified to appreciate and describe the social life of the people than their physical surroundings, and if the results of his great journey are disappointing to the geographer, his See also:
		account of the society of the oasis towns, and of the remarkable men who were then ruling in Hail and Riad, must always possess an absorbing interest as a portrait of Arab life in its freest development.
	encyclopedia.jrank.org /APO_ARN/ARABIA.html (6970 words)

	integration.htm (Site not responding. Last check: 2007-10-22)
		Even at this point, you can still apply some simple estimation rules: if your integrand is positive (or negative) then so should your answer be; if your integrand is less than a well-known function, then its integral will be less than the integral of the well-known function.
		There are various rules for making an estimate for the integrals of the function based on this data.
		This estimate is arrived at (as you might guess from the name) by approximating the area under the graph with trapezia.
	www.maths.ox.ac.uk /prospective-students/undergraduate/single-a-level/integration/html (1894 words)

	maplenotes16.html
		There are other problems which can occur and which also are beyond the scope of this course.
		The Trapezium Rule is (along with Simpson's Rule) one of the more common ways to approximate definite integrals which cannot be found exactly.
		Simpson's Rule is similar to the Trapezium Rule, except that instead of drawing trapezia on the subintervals, it divides each subinterval in two and constructs a
	euclid.ucc.ie /pages/staff/usher/maplenotes16.html (641 words)

	BBC - 16+ SOS Teacher - Maths pure simpson rule
		i am having problems with the simpson rule, trapezoidal and mid ordinate rules can you please explain the formluas to me thanx
		This answer is posted on behalf of Nicholas Dean.
		The area of each interval has been approximated by calculating as a trapezium, which is the average of the two ordinate sides times the base (the rectangle that cuts off a triangle but includes a congruent one from the trapezium).
	www.bbc.co.uk /schools/16/sosteacher/maths/33768.shtml (301 words)

	"Experiments in Undergraduate Mathematics": provisional contents (Site not responding. Last check: 2007-10-22)
		Product rule; quotient rule; composite functions (chain rule); maxima and minima; stationary points; implicitly defined functions.
		Areas under curves; the Riemann approximation and the trapezium rule; indefinite integrals; integration and differentiation; integration of x^n; integrals of common functions; simple rules and patterns.
		Definite integrals; the trapezium rule; parabolic segments; Simpson's rule.
	metric.ma.ic.ac.uk /book/EUM-contents.html (280 words)

	Universal Casio Forum > Trying To Enter Program
		Feb 9 2005, 09:25 AM Hmmm this progrma doesnt seem to be working, the trapezium rule and simpsons rule give similar ansers but integration gives a completely different result.
		Is something wrong in the code posted above ive checked what ive entered a couple of times to check and it appears the same.
		...Somethings wrong with the code for the number of sectors as when the number is increased the trapezium and simpsons values increase.
	www.casiocalc.org /?showtopic=2044 (426 words)

	S:\mma\ch-16\ch16-02.html (Site not responding. Last check: 2007-10-22)
		The following steps define the trapezium rule given by TrapI.
		The last step checks the difference between the result form the trapezium rule and the exact integral,which is possible in this example.
		Try different values for the number of steps n.
	www.keele.ac.uk /depts/ma/mathtech.2nd/ch-16/ch16-02.html (41 words)

	MX3015: Mathematical Computation
		It is in the updated distribution; remember to rename the file before starting work.
		Write a method to evaluate an integral using the trapezium rule.
		package uk.ac.abdn.maths.mx3015.integ; import uk.ac.abdn.maths.mx3015.util.Evaluable; /** * The trapezium rule evaluates an integral by calculating the sum of * the areas of trapeziums.
	www.maths.abdn.ac.uk /~igc/tch/mx3015/integ/integ.html (763 words)

	NEA-0867: FOURACES, MultiGroup Cross-Sections, Resonance Calculation from ENDF/B, KEDAK, UKNDL
		METHOD OF SOLUTION - If the weighting function is simple enough group averaged quantities are computed from the point data and interpolation rule read from the evaluated data library using analytic formulae.
		Otherwise the integrations are performed using the trapezium rule.
		Resonance data are converted into point data using subroutines written primarily for the program CRESO (abstract NEA 0719), then Doppler broadened, and finally group averaged.
	www.nea.fr /abs/html/nea-0867.html (431 words)

	Nai-Chien Huang
		A Fredholm integral equation of the first kind is formulated based on the continuity conditions in the normal components of stress and displacement at the contact interface.
		Trapezium rule of discretization is used for converting the integral equation into a set of linear algebraic equations.
		Two types of remote stresses are considered in this study.
	www.nd.edu /~ame/facultystaff/_Huang,Nai-Chien.html (707 words)

	CATHOLIC ENCYCLOPEDIA: Stole
		A liturgical vestment composed of a strip of material from two to four inches wide and about eighty inches long.
		It has either a uniform width throughout, or is somewhat narrower towards the middle, widening at the ends in the shape of a trapezium or spade.
		A small cross is generally sewed or embroidered on the stole at both ends and in the middle; the cross, however, is prescribed only for the middle, where the priest kisses the stole before putting it on.
	www.newadvent.org /cathen/14301a.htm (1359 words)

	Edinburgh Mathematics Programme (Site not responding. Last check: 2007-10-22)
		Direction fields, Euler’s method, trapezium and Simpson’s rule with extrapolation, Newton-Raphson method.
		Introduction to partial differentiation, directional derivative, differentiation following the motion, differentials and implicit functions.
		Ability to apply Richardson’s Extrapolation to trapezium and Simpson’s rules
	www.maths.ed.ac.uk /guides/Syll/mm2.html (403 words)

	Course Descriptions
		Basic and repeated trapezium and midpoint rules and their error terms.
		Simpson’s rule derived by combining trapezium and midpoint rules.
		The Romberg scheme: extrapolation using the Euler-Maclaurin summation formula.
	www.ma.man.ac.uk /DeptWeb/UGCourses/Syllabus/Level2/MT2282.html (370 words)

	Simpson's Rule for Numerical Integration
		Sharp error bounds for the trapezoidal rule and Simpson's rule.
		On a generalization of a functional equation associated with Simpson's rule.
		An improved rule for qadrature that is closer to the trapezium rule than Simpson's rule
	math.fullerton.edu /mathews/n2003/simpsonsrule/SimpsonsRuleBib/Links/SimpsonsRuleBib_lnk_2.html (390 words)

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