Chapter Simpson s 1/3 Rule of Integration. ( x)

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1 Cper 7. Smpso s / Rule o Iegro Aer redg s per, you sould e le o. derve e ormul or Smpso s / rule o egro,. use Smpso s / rule o solve egrls,. develop e ormul or mulple-segme Smpso s / rule o egro,. use mulple-segme Smpso s / rule o egro o solve egrls, d. derve e rue error ormul or mulple-segme Smpso s / rule. 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 7.A-7.G. 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 Smpso s / rule o egrl ppromo, w mproves upo e ury o e rpezodl rule. Here, we wll dsuss e Smpso s / rule o ppromg egrls o e orm were I s lled e egrd, lower lm o egro upper lm o egro Smpso s / Rule Te rpezodl rule ws sed o ppromg e egrd y rs order polyoml, d e egrg e polyoml over ervl o egro. Smpso s / rule s 7..

2 7.. Cper 7. eeso o Trpezodl rule were e egrd s ppromed y seod order polyoml. Fgure Iegro o uo Meod : Hee I were s seod order polyoml gve y. Coose,,,, d, s e ree pos o e uo o evlue, d. Solvg e ove ree equos or ukows,, d gve

3 Smpso s / Rule o Iegro 7.. Te I Susug vlues o, d gve Se or Smpso / rule, e ervl [ ], s roke o segmes, e segme wd Hee e Smpso s / rule s gve y Se e ove orm s / s ormul, s lled Smpso s / rule. Meod : Smpso s / rule lso e derved y ppromg y seod order polyoml usg Newo s dvded deree polyoml s were

4 7.. Cper 7. Iegrg Newo s dvded deree polyoml gves us Susug vlues o,, d o s equo yelds e sme resul s eore Meod : Oe ould eve use e Lgrge polyoml o derve Smpso s ormul. Noe y meod o ree-po qudr erpolo e used o ompls s sk. I s se, e erpolg uo eomes Iegrg s uo ges

5 Smpso s / Rule o Iegro 7.. Beleve or o, smplyg d org s lrge epresso yelds you e sme resul s eore. Meod : Smpso s / rule lso e derved y e meod o oees. Assume Le e rg-d sde e e epresso or e egrls,, d. Ts mples e rg d sde wll e e epressos or egrls o y ler omo o e ree egrls or geerl seod order polyoml. Now

6 7.. Cper 7. Solvg e ove ree equos or, d gve Ts gves Te egrl rom e rs meod e vewed s e re uder e seod order polyoml, wle e equo rom Meod e vewed s e sum o e res o ree regles. Emple I emp o udersd e mesm o e depolrzo proess uel ell, elero-ke model or med oyge-meol urre o plum ws developed e lorory FAMU. A very smpled model o e reo developed suggess uol relo egrl orm. To d e me requred or % o e oyge o e osumed, e me, s T s gve y T Use Smpso s / rule o d e me requred or % o e oyge o e osumed.

7 Smpso s / Rule o Iegro 7..7 Fd e rue error, E, or pr. Fd e solue relve rue error,, or pr. Soluo T T [ ] 9 s [ ] Te e vlue o e ove egrl s, T...9 s so e rue error s E True Vlue Approme Vlue.9 9.

8 7..8 Cper 7. Te solue relve rue error,, would e e True Vlue True Error.9..7 % Mulple-segme Smpso s / Rule Jus lke mulple-segme rpezodl rule, oe sudvde e ervl [ ], o segmes d pply Smpso s / rule repeedly over every wo segmes. Noe eeds o e eve. Dvde ervl [ ], o equl segmes, so e segme wd s gve y. Now were... Apply Smpso s /rd Rule over e ervl,... Se...,,, e...

9 Smpso s / Rule o Iegro 7..9 [ {... } {... } ] odd eve odd eve Emple I emp o udersd e mesm o e depolrzo proess uel ell, elero-ke model or med oyge-meol urre o plum ws developed e lorory FAMU. A very smpled model o e reo developed suggess uol relo egrl orm. To d e me requred or % o e oyge o e T s s gve y osumed, e me, T Use our-smpso s / Rule o d e me requred or % o e oyge o e osumed. Fd e rue error, E, or pr. Fd e solue relve rue error,, or pr. Soluo T odd eve.....

10 7.. Cper So T odd eve... odd. eve [.. ].

11 Smpso s / Rule o Iegro s Te e vlue o e ove egrl s..7. T s so e rue error s E True Vlue Approme Vlue Te solue relve rue error,, would e e True Error True Vlue % Tle Vlues o Smpso s / Rule or Emple w mulple segmes. Approme Vlue E %

12 7.. Cper 7. Error Mulple-segme Smpso s / rule Te rue error sgle pplo o Smpso s /rd Rule s gve y E ζ, < ζ < 88 I mulple-segme Smpso s / rule, e error s e sum o e errors e pplo o Smpso s / rule. Te error e segmes Smpso s /rd Rule s gve y E ζ, < ζ < 88 ζ 9 E ζ, < ζ < 88 ζ 9 : E ζ, < ζ < 88 ζ 9 : E, < < 88 ζ ζ ζ 9 E ζ, < ζ < 88 Hee, e ol error e mulple-segme Smpso s / rule s ζ 9 E E Te e rue error epresso sds or e our dervve o e uo.

13 Smpso s / Rule o Iegro ζ ζ ζ 9 ζ Te erm s pprome verge vlue o E 9 were ζ, < <. Hee INTEGRATION Top Smpso s / rule Summry Teook oes o Smpso s / rule Mjor Ceml Egeerg Auors Aur Kw, Mel Keels De Novemer 8, We Se p://umerlmeods.eg.us.edu

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