Calculating the Values of Multiple Integrals by Replacing the Integrands Interpolation by Interpolation Polynomial
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1 Jour o omputtos & Modeg vo o -5 ISSN: prt 79-5 oe Scepress Ltd cutg te Vues o Mutpe Itegrs by Repcg te Itegrds Iterpoto by Iterpoto Poyom S Nzrov d bduzzov bstrct Te or des t te costructo o mutdmeso udrture ormus or computto o te mutpe tegrs vues by repcg te tegrds by terpoto poyom We ve proved te udrture ormus correctess Mtemtcs Subect sscto: 5 Keyords: mut-dmeso cubture ormus; terpoto poyoms; mutpe tegrs; te otes coecetsscrz's Itroducto I [-] e cosder te metods o costructg cubture ormus o te Tset Uversty o Iormto Tecooges Tset Tset rtce Io: Receved : pr 7 Revsed : Jue 9 Pubsed oe : September 5
2 cutg te Vues o Mutpe Itegrs bss o te metod o repeted ppcto o deret Smpso d ebysev I ts pper to costruct cubture ormus or te -od tegrs vues ccutg e used metod o repcg o te tegrds by terpotg poyom For ts purpose et s repce te tegrd by te oog poyom: J ere Ω P P Itegrtg e ve: L I smpe eds te d d d Ω P Ω P L d d d P dd d d d d vues ccuto s ot dcut becuse P s poyom I compe res to ccute teс vues t s pproprte to use te VL Rvcev s R-uctos [5] Ts metod determes te pots o tegrto For smpcty et s cosder te -dmeso rectgur rego d some grd es o t ormed by: ; ; ;
3 S Nzrov d bduzzov I ts cse e use te : terpoto ormu usg te odes: R ere 5 Itegrtg t e obt Ω Ω d d Rd d d d d d d J ere R s te remder o terpoto ormu I te st euto troducg te otto d K ; ; 7 e ve d d Rd d d d J Ω re te coecets o umerc tegrto ormus o omogeeous tegrs O ts bss te or [] d troducg te otto
4 cutg te Vues o Mutpe Itegrs [ ] d coecets 7 e reduce to 9 ;!! [ ] d ; No et cosder te spec cses cutg te doube tegrs vues oube tegrs vues ccuto gortm s descrpto Teorem Suppose tt te ucto s deed d cotuous gve rego o tegrto Te te cubture ormu or te doube tegr vues obted by repcg te tegrd by poyom terpoto or d s te orm J d d y [ y y y [ ] y ] Proo I provg teorems e use eutos - I ts cse d respectvey ve te orm dd y ddy J
5 S Nzrov d bduzzov 5!!!! [ ] d ; [ ] d cute te vues o d or d respectvey:!! d d [ ]!! d d I te sme y e ccute te vues O te bss o d bse te d e compute d : d s e s o te vue d tg to ccout : y ; 5 ere Substtutg ts resut e obt Ts resut cocdes t te resut obted by repeted ppcto o te trpezod udrture rue c sos te euvece o te metod o te tegrd terpoto repcg by poyom metod d re-use o udrture ormus Teorem Suppose tt te ucto s deed d cotuous te cosed re o tegrto Te te cubture ormu or te doube tegr
6 cutg te Vues o Mutpe Itegrs vues obted by repcg te tegrd by poyom terpoto or s te orm J y d d [ ] 9 Proo For ts purpose e ccute te d : d vues o or!! d d d!! d d d!! d d I te sme y or e ve Te bsed o te d vuese ccute d Te substtutg te d vues e obt Euto cocdes t te resut obted usg te re-use o Smpso s udrture ormus Ts ct corms te euvece o doube tegrs
7 S Nzrov d bduzzov 7 obted by te repeted ppcto o udrture ormus d metods o repcg te tegrd by terpotg poyom computg Let s rerte d respectvey te orm o compoet s: J S y J T dd 7 y dd 9 To ccute te -od tegrs usg 7 e c derve te correspodg ormus Teorem Let te uctos be deed d cotuous te cosed dom o tegrto Ω Te -dmeso cubture ormus or or d ppromte ccutos retvey te te orm J d d d 9 ere ; J dd d ere Te proo o ts teorem c so be mde by ducto 9 d respectvey cocde t te ormus obted by te repeted ppcto o te udrture trpezod d Smpso d re gve [] Bsed o tese resuts et s me te oog cocusos I e put to 7 e obt cubture ormus o te orm 9 c cocdes t te ormu obted by repeted ppcto o udrture trpeze ormu to ccute vues o mutpe tegrs
8 cutg te Vues o Mutpe Itegrs I e put to 7 e obt ormus or te ppromte ccuto o -od tegrs o te orm c cocdes t te ormu obted by repeted ppcto o te Smpso udrture ormu to compute te vues o mutpe tegrs I e put to 7 e obt te Neto-otes g > order ormus or te ppromte ccuto o -od tegrs d tus provde g order o ccurcy o -od cubture ormus I e put S S S to 7 e get te -od composte cubture ormus or -od tegrs Te essece o mes tt or ec e use deret Neto-otes ormus Ts s o terest to te costructo o -dmeso cubture ormus Teorem Let be deed d cotuous te cosed re o tegrto Te te cubture ormu or te ppromte ccuto o vues by repcg te tegrd terpoto by poyom obted e s te orm dd d y y {[ ] [ y y y y y y y y y y ] 9[ y y y y ]} ere 9
9 S Nzrov d bduzzov 9 Proo For ts purpose ccordce t prgrp e derve cubture ormus to ccute vues or doube tegrs or I ts cse te d vues ccordg to te te orm!!!!! ] [ d d d!!!!!! 5!!!!!!!! d d d d d d I te sme y or e ve Let s rerte te ormu te orm o: 9 d d d d ere!!!!!!
10 cutg te Vues o Mutpe Itegrs Teorem 5 Let be deed d cotuous -dmeso tegrto rego Ω Te cubture ormu or te ppromte computto o te vues o te tegrd by terpoto poyom obted or s te orm: d d d! ere!! d d d 5 Proo I te eutos to ccute te vues o doube tegrs 7 d respectvey d produced by d d te vues o -od tegrs 9 d respectvey d produced by и d to ccute te vues o -od tegrs s re useu e te sze o te rego s very sm I te dmesos re rge te to crese te ccurcy o te -od cubture ormus o ec ] e dvde te system uder te [ [ ] [ ] [ ] [ ] tegrs d ppy to ec te pproprte cubture ormus To uderstd ts pproc uy e rst cosder te doube tegrs ere te to ] d ] [ [ terpotos correspod to drectos og teod o es Te [ ] tervs re dvded by d te [ ] terv by prts Net or e ve b
11 S Nzrov d bduzzov d or b y 7 Tg to ccout 5 d te 7 d c be respectvey rertte s d d d d 9 d d d d α 9 Resog by ogy d 9 te ormus 9 d respectvey re rertte s oog: d d d d d d d d d d d d d d d d d d It soud be oted tt te ccuto o mutpe tegrs be preset uder te sum d e - etc Note tt ts rue sce s voted For crty e preset beo te otes coecets Tbe
12 cutg te Vues o Mutpe Itegrs ccordg to te vues o umbers stdg ros d coums see Tbe te otes coecets re cresg rpdy t te cresg o vues d sce become egtve Tereore e ve deveoped progrm Mpe mtemtc system or te coecets d costructed sure Neto - otes ormus ccutg We o gve udrture ormus costructed by usg ts Mpe procedure I Tbe or coveece ccordg to [] te otes recordg rte or ec Ĥ s preseted te orm o rctos: N s cer tt N egtve t commo ctor N It ccordg Tbe t rge vueste resuts my be No te vue o ts tegr s ccuted ppromtey by te Neto-otes e seve ordtes Empe Let s ccute Y y ddy Tbe
13 S Nzrov d bduzzov otuto o tbe Vue obects Te ect vue o te tegr s te oog: I T d y ydy ; y Let s dr up tbe o vues Tbe ssumg tt
14 cutg te Vues o Mutpe Itegrs Tbe : cutg vues y y y y Σ 5997 No bsed o ts tbe e ccute te vue o doube tegr: I пр despte te ct tt te vues I T precso d I пр ppromte re cose eoug og t ts e ote tt te o otes coecet re egtve rge bsoute vue tt re very deret rom ec oter Tereore te Neto - otes s ot recommeded or rge
15 S Nzrov d bduzzov 5 Isted e or or ec tegrto terv soud be dvded to prts d use te pproprte Neto-otes ec o tem ocusos Tus e ve costructed te mut-dmeso cubture ormus to ppromte te vues o mutpe tegrs by repcg te tegrds by terpoto poyom c c be esy mpemeted o computer Reereces [] S Nzrov gortm or computg te mutpe tegrs vues t combto o Mote ro d R-uctos Iss g d gue Mtemtcs St scetc Tset [] S Nzrov ppromte ccuto o tegrs by te bre oedge o udrture Smpso ormus re-use g d gue Mtemtcs St scetc Tset [] SNzrov Te metod o re-use o udrture ormus or trpezums d rectges to determe te vues o mutpe tegrs g d gue Mtemtcs St scetc Tset [] BP emdovc d I Mro Foudtos o omputto Mtemtcs N: Scece 9 p [5] VL Rvcev Teory o R - uctos d some ppctos Kev Nuov um 95 p 5
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