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
Guss Qudrture Rule o Integrtion http://numerilmethods.eng.us.edu
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 http://numerilmethods.eng.us.edu
Two-Point Gussin Qudrture Rule 4 http://numerilmethods.eng.us.edu
http://numerilmethods.eng.us.edu 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
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 http://numerilmethods.eng.us.edu
http://numerilmethods.eng.us.edu 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 0 4 4 0 ) 4 4 4 0
http://numerilmethods.eng.us.edu 8 Bsis o the Gussin Qudrture Rule It ollows tht ) ) 0 0 )d Equting Equtions the two previous two epressions yield ) 4 4 4 0 ) ) 0 0 ) ) ) ) 0
http://numerilmethods.eng.us.edu 9 Bsis o the Gussin Qudrture Rule Sine the onstnts 0,,, re ritrry 4 4 4
http://numerilmethods.eng.us.edu 0 Bsis o Guss Qudrture The previous our simultneous nonliner Equtions hve only one eptle solution,
http://numerilmethods.eng.us.edu Bsis o Guss Qudrture Hene Two-Point Gussin Qudrture Rule ) ) ) d
Higher Point Gussin Qudrture Formuls http://numerilmethods.eng.us.edu
http://numerilmethods.eng.us.edu 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 5 5 4 4 0 Generl n-point rules would pproimte the integrl )....... ) ) )d n n integrting ith order polynomil
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.000000000.000000000 0.555555556 0.888888889 0.555555556 4 0.47854845 0.654555 0.654555 4 0.47854845 Funtion Arguments -0.5775069 0.5775069-0.774596669 0.000000000 0.774596669-0.866-0.998044 0.998044 4 0.866 4 http://numerilmethods.eng.us.edu
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 5 0.696885 0.47868670 0.568888889 4 0.47868670 5 0.696885 6 0.7449 0.607657 0.467995 4 0.467995 5 0.607657 6 0.7449 Funtion Arguments -0.90679846-0.584690 0.000000000 4 0.584690 5 0.90679846-0.946954-0.660986-0.869860 4 0.869860 5 0.660986 6 0.946954 5 http://numerilmethods.eng.us.edu
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 http://numerilmethods.eng.us.edu
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 http://numerilmethods.eng.us.edu
Emple For n integrl Rule. )d, derive the one-point Gussin Qudrture Solution The one-point Gussin Qudrture Rule is )d ) 8 http://numerilmethods.eng.us.edu
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 http://numerilmethods.eng.us.edu
Solution It ollows tht ) ) d Equting Equtions, the two previous two epressions yield 0 ) 0 ) 0 ) ) 0 0 http://numerilmethods.eng.us.edu
Bsis o the Gussin Qudrture Rule Sine the onstnts 0, nd re ritrry giving http://numerilmethods.eng.us.edu
Solution Hene One-Point Gussin Qudrture Rule ) d ) ) http://numerilmethods.eng.us.edu
Emple ) Use two-point Guss Qudrture Rule to pproimte the distne overed y roket rom t8 to t0 s given y 0 40000 000ln 9. 8t dt 8 40000 00t ) Find the true error, E t or prt ). ) Also, ind the solute reltive true error, or prt ). http://numerilmethods.eng.us.edu
Solution First, hnge the limits o integrtion rom [8,0] to [-,] y previous reltions s ollows 0 t )dt 0 8 0 8 0 8 8 d 9) d 4 http://numerilmethods.eng.us.edu
Solution ont) Net, get weighting tors nd untion rgument vlues rom Tle or the two point rule,. 000000000 0. 5775069. 000000000 0. 5775069 5 http://numerilmethods.eng.us.edu
Solution ont.) Now we n use the Guss Qudrture ormul 9) d 9) 9) 0. 57750 ) 9) 0. 57750 ) 9). 6495 ) 5. 5085 ) 96. 87 ) 708. 48) 058.44 m 6 http://numerilmethods.eng.us.edu
Solution ont) sine 40000. 6495 ) 000ln 9. 8. 6495 ) 40000 00. 6495 ) 96.87 40000 5. 5085 ) 000ln 9. 8 5. 5085 ) 40000 00 5. 5085 ) 708.48 7 http://numerilmethods.eng.us.edu
Solution ont) ) The true error, t ) E t E, is True Vlue Approimte Vlue 06.4 058.44.9000 m The solute reltive true error, t, is Et vlue 06.4m) 06. 4 058. 44 t 00% 06. 4 8 0.06% http://numerilmethods.eng.us.edu
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 http://numerilmethods.eng.us.edu/topis/guss_qu drture.html
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