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 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 ) Use two-point 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, multiple-hoie tests, worksheets in MATLAB, MATHEMATICA, MthCd nd MAPLE, logs, relted physil prolems, plese visit drture.html

30 THE END