Gauss Quadrature Rule of Integration


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1 Guss Qudrture Rule o Integrtion Mjor: All 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 TwoPoint 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 twopoint 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 twopoint 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 TwoPoint Gussin Qudrture Rule ) ) ) d
12 Higher Point Gussin Qudrture Formuls
13 Higher Point Gussin Qudrture Formuls ) ) ) ) d is lled the threepoint Guss Qudrture Rule. The oeiients,, nd, nd the untionl rguments,, nd re lulted y ssuming the ormul gives et epressions or ) d Generl npoint rules would pproimte the integrl ) ) ) )d n n integrting ith order polynomil
14 Arguments nd Weighing Ftors or npoint Guss Qudrture Formuls In hndooks, oeiients nd rguments given or npoint 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 npoint 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 npoint 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 npoint 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 onepoint Gussin Qudrture Solution The onepoint 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 OnePoint Gussin Qudrture Rule ) d ) )
23 Emple ) Use twopoint Guss Qudrture Rule to pproimte the distne overed y roket rom t8 to t0 s given y ln 9. 8t dt t ) Find the true error, E t or prt ). ) Also, ind the solute reltive true error, or prt ).
24 Solution First, hnge the limits o integrtion rom [8,0] to [,] y previous reltions s ollows 0 t )dt d 9) 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 9) d 9) 9) ) 9) ) 9) ) ) ) ) m 6
27 Solution ont) sine ) 000ln ) ) ) 000ln ) )
28 Solution ont) ) The true error, t ) E t E, is True Vlue Approimte Vlue m The solute reltive true error, t, is Et vlue 06.4m) t 00% %
29 Additionl Resoures For ll resoures on this topi suh s digitl udiovisul letures, primers, tetook hpters, multiplehoie 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 Computer 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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