Gauss Quadrature Rule of Integration
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1 Guss Qudrture Rule o Integrtion Computer Engineering Mjors Authors: Autr Kw, Chrlie Brker Trnsorming Numeril Methods Edution or STEM Undergrdutes /0/00
2 Guss Qudrture Rule o Integrtion
3 Wht is Integrtion? Integrtion The proess o mesuring the re under urve. y )d ) I )d Where: ) is the integrnd lower limit o integrtion upper limit o integrtion
4 Two-Point Gussin Qudrture Rule 4
5 5 Bsis o the Gussin Qudrture Rule Previously, the Trpezoidl Rule ws developed y the method o undetermined oeiients. The result o tht development is summrized elow. ) ) ) ) ) d
6 Bsis o the Gussin Qudrture Rule The two-point Guss Qudrture Rule is n etension o the Trpezoidl Rule pproimtion where the rguments o the untion re not predetermined s nd ut s unknowns nd. In the two-point Guss Qudrture Rule, the integrl is pproimted s I )d ) ) 6
7 7 Bsis o the Gussin Qudrture Rule The our unknowns,, nd re ound y ssuming tht the ormul gives et results or integrting generl third order polynomil,. ) 0 Hene ) d )d )
8 8 Bsis o the Gussin Qudrture Rule It ollows tht ) ) 0 0 )d Equting Equtions the two previous two epressions yield ) ) ) 0 0 ) ) ) ) 0
9 9 Bsis o the Gussin Qudrture Rule Sine the onstnts 0,,, re ritrry 4 4 4
10 0 Bsis o Guss Qudrture The previous our simultneous nonliner Equtions hve only one eptle solution,
11 Bsis o Guss Qudrture Hene Two-Point Gussin Qudrture Rule ) ) ) d
12 Higher Point Gussin Qudrture Formuls
13 Higher Point Gussin Qudrture Formuls ) ) ) ) d is lled the three-point Guss Qudrture Rule. The oeiients,, nd, nd the untionl rguments,, nd re lulted y ssuming the ormul gives et epressions or ) d Generl n-point rules would pproimte the integrl ) ) ) )d n n integrting ith order polynomil
14 Arguments nd Weighing Ftors or n-point Guss Qudrture Formuls In hndooks, oeiients nd rguments given or n-point Guss Qudrture Rule re given or integrls g )d n i s shown in Tle. i g i ) Tle : Weighting tors nd untion rguments used in Guss Qudrture Formuls. Points Weighting Ftors Funtion Arguments
15 Arguments nd Weighing Ftors or n-point Guss Qudrture Formuls Tle ont.) : Weighting tors nd untion rguments used in Guss Qudrture Formuls. Points Weighting Ftors Funtion Arguments
16 Arguments nd Weighing Ftors or n-point Guss Qudrture So i the tle is given or g Formuls )d integrls, how does one solve )d? The nswer lies in tht ny integrl with limits o [, ] n e onverted into n integrl with limits [,] mt Let I, then t I then t, Suh tht: 6 m
17 Arguments nd Weighing Ftors or n-point Guss Qudrture Formuls Then Hene t d dt Sustituting our vlues o, nd d into the integrl gives us ) d t dt 7
18 Emple For n integrl Rule. )d, derive the one-point Gussin Qudrture Solution The one-point Gussin Qudrture Rule is )d ) 8
19 Solution The two unknowns, nd re ound y ssuming tht the ormul gives et results or integrting generl irst order polynomil, ) 0. ) d ) 0 d 0 0 ) 9
20 Solution It ollows tht ) ) d Equting Equtions, the two previous two epressions yield 0 ) 0 ) 0 ) ) 0 0
21 Bsis o the Gussin Qudrture Rule Sine the onstnts 0, nd re ritrry giving
22 Solution Hene One-Point Gussin Qudrture Rule ) d ) )
23 Emple Humn vision hs the remrkle ility to iner D shpes rom D imges. The intriguing question is: n we replite some o these ilities on omputer? Yes, it n e done nd to do this, integrtion o vetor ields is required. The ollowing integrl needs to integrted. 00 I ) d where ) 0, 0 < < , 7 < < Use two-point Guss Qudrture Rule to ind the vlue o the integrl. Also, ind the solute reltive true error ,
24 Solution First, hnge the limits o integrtion rom [0,00] to [-,] y previous reltions s ollows ) d d ) 50 d 4
25 Solution ont). Net, get weighting tors nd untion rgument vlues rom Tle or the two point rule,
26 Solution ont.) Now we n use the Guss Qudrture ormul ) [ 50 50) 50 50) ] d [ ) 50) ) 50) ] [.) ) ] [ 0) 0.049) ]
27 Solution ont) sine.) ) ) ) )
28 Solution ont) ) True error is True Vlue Approimte E t Vlue ) The solute reltive true error,, is Et vlue 60.79) t t 00% 9.7% 8
29 Additionl Resoures For ll resoures on this topi suh s digitl udiovisul letures, primers, tetook hpters, multiple-hoie tests, worksheets in MATLAB, MATHEMATICA, MthCd nd MAPLE, logs, relted physil prolems, plese visit drture.html
30 THE END
Gauss Quadrature Rule of Integration
Guss Qudrture Rule o Integrtion Mjor: All Engineering Mjors Authors: Autr Kw, Chrlie Brker http://numerilmethods.eng.us.edu Trnsorming Numeril Methods Edution or STEM Undergrdutes /0/00 http://numerilmethods.eng.us.edu
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