Chapter Simpson s 1/3 Rule of Integration. ( x)
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1 Cpter 7. Smpso s / Rule o Itegrto Ater redg ts pter, you sould e le to. derve te ormul or Smpso s / rule o tegrto,. use Smpso s / rule t to solve tegrls,. develop te ormul or multple-segmet Smpso s / rule o tegrto,. use multple-segmet Smpso s / rule o tegrto to solve tegrls, d. derve te true error ormul or multple-segmet Smpso s / rule. Wt s tegrto? Itegrto s te proess o mesurg te re uder uto plotted o grp. Wy would we wt to tegrte uto? Amog te most ommo emples re dg te veloty o ody rom elerto uto, d dsplemet o ody rom veloty uto. Trougout my egeerg elds, tere re wt sometmes seems lke outless ppltos or tegrl lulus. You red out some o tese ppltos Cpters 7.A-7.G. Sometmes, te evluto o epressos volvg tese tegrls eome dutg, ot determte. For ts reso, wde vrety o umerl metods s ee developed to smply te tegrl. Here, we wll dsuss Smpso s / rule o tegrl ppromto, w mproves upo te ury o te trpezodl rule. Here, we wll dsuss te Smpso s / rule o ppromtg tegrls o te orm were I d s lled te tegrd, lower lmt o tegrto upper lmt o tegrto Smpso s / Rule Te trpezodl rule ws sed o ppromtg te tegrd y rst order polyoml, d te tegrtg te polyoml over tervl o tegrto. Smpso s / rule s 7..
2 7.. Cpter 7. eteso o Trpezodl rule were te tegrd s ppromted y seod order polyoml. Fgure Itegrto o uto Metod : Hee I d d were s seod order polyoml gve y. Coose,,,, d, s te tree pots o te uto to evlute, d. Solvg te ove tree equtos or ukows,, d gve
3 Smpso s / Rule o Itegrto 7.. Te d I d Susttutg vlues o, d gve d Se or Smpso / rule, te tervl [ ], s roke to segmets, te segmet wdt Hee te Smpso s / rule s gve y d Se te ove orm s / ts ormul, t s lled Smpso s / rule. Metod : Smpso s / rule lso e derved y ppromtg y seod order polyoml usg Newto s dvded deree polyoml s were
4 7.. Cpter 7. Itegrtg Newto s dvded deree polyoml gves us d d d Susttutg vlues o,, d to ts equto yelds te sme result s eore d Metod : Oe ould eve use te Lgrge polyoml to derve Smpso s ormul. Note y metod o tree-pot qudrt terpolto e used to ompls ts tsk. I ts se, te terpoltg uto eomes Itegrtg ts uto gets
5 Smpso s / Rule o Itegrto 7.. d Beleve t or ot, smplyg d torg ts lrge epresso yelds you te sme result s eore d. Metod : Smpso s / rule lso e derved y te metod o oeets. Assume d Let te rgt-d sde e et epresso or te tegrls, d, d d d. Ts mples tt te rgt d sde wll e et epressos or tegrls o y ler omto o te tree tegrls or geerl seod order polyoml. Now d
6 7.. Cpter 7. d d Solvg te ove tree equtos or, d gve Ts gves d Te tegrl rom te rst metod d d e vewed s te re uder te seod order polyoml, wle te equto rom Metod d e vewed s te sum o te res o tree retgles. Emple Te dste overed y roket meters rom 8 t s to t s s gve y l dt t t Use Smpso s / rule to d te ppromte vlue o. Fd te true error, t E. Fd te solute reltve true error, t.
7 Smpso s / Rule o Itegrto 7..7 Soluto 8 9 t l 9. 8t t 8 l m 8 l m 9 l m 9 8 [ 8 9 ] [ ].7 m Te et vlue o te ove tegrl s l 9.8t dt t 8. m So te true error s E t True Vlue Appromte Vlue m Te solute reltve true error s True Error t True Vlue.8.
8 7..8 Cpter 7..9% Multple-segmet Smpso s / Rule Just lke multple-segmet trpezodl rule, oe sudvde te tervl [ ], to segmets d pply Smpso s / rule repetedly over every two segmets. Note tt eeds to e eve. Dvde tervl [ ], to equl segmets, so tt te segmet wdt s gve y. Now d d were d d d d d... Apply Smpso s /rd Rule over e tervl,... d Se...,,, te... d { } { } [ ]......
9 Smpso s / Rule o Itegrto 7..9 odd eve Emple d odd eve Use -segmet Smpso s / rule to ppromte te dste overed y roket meters rom t 8 s to t s s gve y l 9.8t dt t 8 Use our segmet Smpso s /rd Rule to estmte. Fd te true error, E t or prt. Fd te solute reltve true error, t or prt. Soluto: Usg segmet Smpso s / rule, t t t odd eve 8 8. t l 9. 8t t t So t 8 8 l m 8 t 8..
10 7.. Cpter 7.. l 9.8..m. t l m 9 t 9... l m. t t l m t t t odd eve 8 8 t t odd eve [ 8 t t t ] [ ] m Te et vlue o te ove tegrl s [ ] l 9.8t dt t 8. m So te true error s True Vlue Appromte Vlue E t E t... m t
11 Smpso s / Rule o Itegrto 7.. Te solute reltve true error s True Error t True Vlue...7% Tle Vlues o Smpso s / rule or Emple wt multple-segmets Appromte Vlue Et t %.7%.%.%.% Error Multple-segmet Smpso s / rule Te true error sgle pplto o Smpso s /rd Rule s gve y E t, < < 88 I multple-segmet Smpso s / rule, te error s te sum o te errors e pplto o Smpso s / rule. Te error te segmets Smpso s /rd Rule s gve y E, < < 88 9 E, < < 88 9 : E, < < 88 9 : Te te true error epresso stds or te ourt dervtve o te uto.
12 7.. Cpter 7., 88 < < E 9 E < <, 88 Hee, te totl error te multple-segmet Smpso s / rule s 9 E t E 9 9, 8 Te term s ppromte verge vlue o < <,. Hee 8 E t were INTEGRATION Top Smpso s / rule Summry Tetook otes o Smpso s / rule Mjor Geerl Egeerg Autors Autr Kw, Mel Ketelts
13 Smpso s / Rule o Itegrto 7.. Dte Deemer, 7 We Ste ttp://umerlmetods.eg.us.edu
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Numerl Metods or Eg [ENGR 9] [Lyes KADEM 7] CHAPTER VI Numerl Itegrto Tops - Rem sums - Trpezodl rule - Smpso s rule - Rrdso s etrpolto - Guss qudrture rule Mtemtlly, tegrto s just dg te re uder urve rom
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