Chapter Trapezoidal Rule of Integration

Transcription

1 Cper 7 Trpezodl Rule o Iegro Aer redg s per, you sould e le o: derve e rpezodl rule o egro, use e rpezodl rule o egro o solve prolems, derve e mulple-segme rpezodl rule o egro, 4 use e mulple-segme rpezodl rule o egro o solve prolems, d 5 derve e ormul or e rue error e mulple-segme rpezodl rule o egro W s egro? Iegro s e proess o mesurg e re uder uo ploed o grp Wy would we w o egre uo? Amog e mos ommo emples re dg e veloy o ody rom elero uo, d dspleme o ody rom veloy uo Trougou my egeerg elds, ere re w somemes seems lke ouless pplos or egrl lulus You red ou some o ese pplos Cpers 7A-7G Somemes, e evluo o epressos volvg ese egrls eome dug, o deerme For s reso, wde vrey o umerl meods s ee developed o smply e egrl Here, we wll dsuss e rpezodl rule o ppromg egrls o e orm were I d s lled e egrd, lower lm o egro upper lm o egro W s e rpezodl rule? Te rpezodl rule s sed o e Newo-Coes ormul oe ppromes e egrd y order polyoml, e e egrl o e uo s ppromed y 7

2 7 Cper 7 e egrl o order polyoml Iegrg polyomls s smple d s sed o e lulus ormul Fgure Iegro o uo d, So we w o pprome e egrl I d o d e vlue o e ove egrl, oe ssumes were 4 were s order polyoml Te rpezodl rule ssumes, s, ppromg e egrl y ler polyoml srg le, d d Dervo o e Trpezodl Rule Meod : Derved rom Clulus d d d 5

3 Trpezodl Rule 7 Bu w s d? Now oe ooses,, d, s e wo pos o pprome y srg le rom o, 6 7 Solvg e ove wo equos or d, 8 Hee rom Equo 5, d 8 9 Meod : Also Derved rom Clulus lso e ppromed y usg Newo s dvded deree polyoml s Hee d d d

4 74 Cper 7 Ts gves e sme resul s Equo euse ey re jus dere orms o wrg e sme polyoml Meod : Derved rom Geomery Te rpezodl rule lso e derved rom geomery Look Fgure Te re uder e urve s e re o rpezod Te egrl rpezod o Are d Sum o leg o prllel sdesperpedulr dse ewee prllel sdes Fgure Geomer represeo o rpezodl rule Meod 4: Derved rom Meod o Coees Te rpezodl rule lso e derved y e meod o oees Te ormul d were

5 Trpezodl Rule 75 Fgure Are y meod o oees Te erpreo s s evlued pos d, d e uo evluo s gve weg o Geomerlly, Equo s looked s e re o rpezod, wle Equo s vewed s e sum o e re o wo regles, s sow Fgure How oe derve e rpezodl rule y e meod o oees? Assume d 4 Le e rg d sde e e epresso or egrls o d d d, s, e ormul wll e lso e e or ler omos o d, s, or d 5 d 6 Solvg e ove wo equos gves 7 Hee

6 76 Cper 7 d 8 Meod 5: Aoer ppro o e Meod o Coees Te rpezodl rule lso e derved y e meod o oees y oer ppro d Assume d 9 Le e rg d sde e e or egrls o e orm d So d Bu we w d o gve e sme resul s Equo or d Hee rom Equos d, Se d re rrry or geerl srg le Ag, solvg e ove wo equos gves 4

7 Trpezodl Rule 77 Tereore d 5 Emple Hum vso s e remrkle ly o er D spes rom D mges Te rgug queso s: we reple some o ese les o ompuer? Yes, e doe d o do s, egro o veor elds s requred Te ollowg egrl eeds o egred Were, I d, < < , 7 < < , Use sgle segme Trpezodl rule o d e vlue o e egrl Fd e rue error, E, or pr Fd e solue relve rue error or pr 7 Soluo I, were, < < , 7 < < 7 7 I , Te e vlue o e ove egrl s oud usg Mple or lulg e rue error d relve rue error

8 78 Cper 7 I d 679 so e rue error s E True Vlue Approme Vlue Te solue relve rue error,, would e e True Error % True Vlue % 859 % Mulple-Segme Trpezodl Rule I Emple, e rue error usg sgle segme rpezodl rule ws lrge We dvde e ervl [,] o [,5] d [ 5,] ervls d pply e rpezodl rule over e segme Hee d 5 d d d d Te rue error, E s E

9 Trpezodl Rule 79 Te rue error ow s redued rom 64 6 o 745 Eedg s proedure o dvdg ], [ o equl segmes d pplyg e rpezodl rule over e segme, e sum o e resuls oed or e segme s e pprome vlue o e egrl Dvde o equl segmes s sow Fgure 4 Te e wd o e segme s 6 Te egrl I e roke o egrls s d I d d d d 7 Fgure 4 Mulple 4 segme rpezodl rule Applyg rpezodl rule Equo 7 o e segme gves [ ] d [ ] [ ] [ ]

10 7 Cper 7 8 Emple Hum vso s e remrkle ly o er D spes rom D mges Te rgug queso s: we reple some o ese les o ompuer? Yes, e doe d o do s, egro o veor elds s requred Te ollowg egrl eeds o egred d I were 7, , < < < <, Use -segme Trpezodl rule o d e vlue o e egrl Fd e rue error, E, or pr Fd e solue relve rue error,,or pr Soluo I 5

11 Trpezodl Rule 7 Se I, < < , 7 < < [ 5 ] [ ] , Te e vlue o e ove egrl s oud usg Mple or lulg e rue error d relve rue error I d 679 so e rue error s E True Vlue Approme Vlue Te solue relve rue error,, would e e True Error % True Vlue % %

12 7 Cper 7 Tle Vlues oed usg mulple-segme Trpezodl rule or, < < , 7, 7 < < Vlue E % % Emple Use e mulple-segme rpezodl rule o d e re uder e urve e rom o Soluo Usg wo segmes, we ge 5 e e 6 e I 5 [ 5 ] 4 [ 9 6]

13 Trpezodl Rule 7 So w s e rue vlue o s egrl? d 4659 e Mkg e solue relve rue error % Wy s e rue vlue so r wy rom e pprome vlues? Jus ke look Fgure 5 As you see, e re uder e rpezods ye, ey relly look lke rgles ow overs smll poro o e re uder e urve As we dd more segmes, e ppromed vlue qukly pproes e rue vlue Fgure 5 -segme rpezodl rule ppromo Tle Vlues oed usg mulple-segme rpezodl rule or d e Approme E Vlue % % % % % % %

14 74 Cper 7 Emple 4 Use mulple-segme rpezodl rule o d I d Soluo We o use e rpezodl rule or s egrl, s e vlue o e egrd s e However, s kow dsouy urve wll o ge e re uder We ssume y vlue or e uo Te lgorm o dee e uo so we use e mulple-segme rpezodl rule s gve elow Fuo I Te I Te ^ 5 Ed Fuo Bslly, we re jus ssgg e uo vlue o zero Everywere else, e uo s ouous Ts mes e rue vlue o our egrl wll e jus rue Le s see w ppes usg e mulple-segme rpezodl rule Usg wo segmes, we ge 77 I [ ] 4 [ 77] 4 56 So w s e rue vlue o s egrl? d 884 Tus mkg e solue relve rue error

15 Trpezodl Rule % Tle Vlues oed usg mulple-segme rpezodl rule or Approme E Vlue % % % % % % % % % % % % d Error Mulple-segme Trpezodl Rule Te rue error or sgle segme Trpezodl rule s gve y E "ζ, < ζ < Were ζ s some po [, ] W s e error e e mulple-segme rpezodl rule? I wll e smply e sum o e errors rom e segme, were e error e segme s o e sgle segme rpezodl rule Te error e segme s [ ] E " ζ, < ζ < " ζ [ ] E " ζ, < ζ < " ζ [ ] E " ζ, < ζ <

16 76 Cper 7 " ζ [{ } { } ] E " ζ, < ζ < " ζ [ { } ] E " ζ, < ζ < " ζ Hee e ol error e mulple-segme rpezodl rule s E E " ζ " ζ " ζ " ζ Te erm dervve ", < < Hee s pprome verge vlue o e seod " ζ E I Tle 4, e pprome vlue o e egrl 4 l 98 d 4 8 s gve s uo o e umer o segmes You vsulze s e umer o segmes re douled, e rue error ges ppromely qurered

17 Trpezodl Rule 77 Tle 4 Vlues oed usg mulple-segme rpezodl rule or 4 l 98 d 4 8 Approme E % Vlue % For emple, or e -segme rpezodl rule, e rue error s -5, d qurer o error s -55 T s lose o e rue error o -48 or e 4-segme rpezodl rule C you swer e queso wy s e rue error o ely -55? How does s ormo elp us umerl egro? You wll d ou s orms e ss o Romerg egro sed o e rpezodl rule, were we use e rgume rue error ges ppromely qurered we e umer o segmes s douled Romerg egro sed o e rpezodl rule s ompuolly more ee usg e rpezodl rule y sel developg uom egro seme INTEGRATION Top Trpezodl Rule Summry Teook oes o e rpezodl rule o egro Mjor Compuer Egeerg Auors Aur Kw, Mel Keels De Novemer, We Se p://umerlmeodsegusedu