Mathematically, integration is just finding the area under a curve from one point to another. It is b
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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](/docs-images/90/103075936/images/1-0.jpg "rom oe pot to oter It s d represeted y, were te symol s tegrl sg, te umers d re te lower d upper lmts o tegrto, respetvely, te uto s te tegrd o te tegrl, d s te vrle o tegrto Fgure represets grpl")
1 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 oe pot to oter It s d represeted y, were te symol s tegrl sg, te umers d re te lower d upper lmts o tegrto, respetvely, te uto s te tegrd o te tegrl, d s te vrle o tegrto Fgure represets grpl demostrto o te oept Wy re we terested tegrto: euse most equtos pyss re deretl equtos tt must e tegrted to d te solutos Furtermore, some pysl quttes e oted y tegrto emple: dsplemet rom veloty Te prolem s tt sometmes tegrtg lytlly some utos esly eome lorous For ts reso, wde vrety o umerl metods ve ee developed to d te tegrl I Rem Sums Fgure- Itegrto Let e deed o te losed tervl [, ], d let e rtrry prtto o [, ] su s: < < < < - <, were s te legt o te t sutervl I s y pot te t sutervl, te te sum Δ, Numerl Itegrto 8
2 Numerl Metods or Eg [ENGR 9] [Lyes KADEM 7] - - s lled Rem Sum o te uto or te prtto o te tervl [, ] For gve prtto, te legt o te logest sutervl s lled te orm o te prtto It s deoted y te orm o Te ollowg lmt s used to dee te dete tegrl: lm Δ Δ I Ts lmt ests d oly or y postve umer ε, tere ests postve umer δ su tt or every prtto o [, ] wt < δ, t ollows tt I Δ < ε or y oe o te umers te t sutervl o I te lmt o Rem Sum o ests, te te uto s sd to e tegrle over [, ] d tt te Rem Sums o o [, ] ppro te umer I Emple lm Δ I, Δ Were I d Fd te re o te rego etwee te prol y d te -s o te tervl [, ] Use Rem s Sum wt our prttos TRAPEZOIDAL RULE Trpezodl rule s sed o te Newto-Cotes ormul tt we ppromte te tegrd y t order polyoml, te te tegrl o te uto s ppromted y te tegrl o tt t order polyoml d, Numerl Itegrto 8
3 Numerl Metods or Eg [ENGR 9] [Lyes KADEM 7] So we wt to ppromte te tegrl d I to d te vlue o te ove tegrl, we wrte our uto uder polyoml orm: were were s order polyoml Trpezodl rule ssumes t, tt s, te re uder te ler polyoml strgt le, d d DERIVATION OF THE TRAPEZOIDAL RULE We ve: d d d But wt s d? Now we oose, d s te two pots to ppromte y strgt le rom to,,, Solvg te ove two equtos or d, Hee we get, d d Numerl Itegrto 8
4 Numerl Metods or Eg [ENGR 9] [Lyes KADEM 7] Emple Te vertl dste overed y roket rom t 8 to t seods s gve y 8 l 98t dt t Use sgle segmet Trpezodl rule to d te dste overed Fd te true error, E t or prt Fd te solute reltve true error or prt Multple-segmet Trpezodl Rule: Oe wy to rese te ury o te trpezodl rule s to rese te umer o segmets etwee d So ts proedure, we wll dvde [, ] to equl segmets d pply te Trpezodl rule over e segmet, te sum o te results oted or e segmet s te ppromte vlue o te tegrl Dvde to equl segmets s sow te gure elow Te te wdt o e segmet s Te tegrl Ι e roke to tegrls s I d d d d d Fgure- Multple-segmet Trpezodl rule Numerl Itegrto 8
5 Numerl Metods or Eg [ENGR 9] [Lyes KADEM 7] Applyg Trpezodl rule o e segmet gves: d [ ] [ ] d Emple Te vertl dste overed y roket rom t 8 to t seods s gve y 8 l 98t dt t Use two-segmet Trpezodl rule to d te dste overed Fd te true error, E t or prt Fd te solute reltve true error or prt Wy resg te umer o segmets To llustrte te mporte o resg te umer o segmets te Trpezodl rule, let us osder te ollowg tegrl: I d e Numerl Itegrto 8
6 Numerl Metods or Eg [ENGR 9] [Lyes KADEM 7] Te ollowg tle represets te vrto te solute d reltve error wt te umer o segmets used Note tt wt smll umer o segmets, te error s very g Appromte Vlue % 9 79% % % % 7 9% 8 % E t t Error Multple-segmet Trpezodl Rule Te true error or sgle segmet Trpezodl rule s gve y E t "ζ, < ζ < [,] were ζ s some pot Wt s te error, te, te multple-segmet Trpezodl rule? It wll e smply te sum o te errors rom e segmet, were te error e segmet s tt o te sgle segmet Trpezodl rule Te error e segmet s [ ] E " ζ < ζ < " ζ [ ] E " ζ < ζ < " ζ [ { } ] E " ζ < ζ < " ζ Hee te totl error multple-segmet Trpezodl rule s E t E Numerl Itegrto 8
7 Numerl Metods or Eg [ENGR 9] [Lyes KADEM 7] Te term Hee " ζ " ζ " ζ " ζ ", < < s ppromte verge vlue o te seod dervtve E t " ζ SIMPSON S / RD RULE Trpezodl rule ws sed o ppromtg te tegrd y rst order polyoml, d te tegrtg te polyoml te tervl o tegrto Smpso s /rd rule s eteso o Trpezodl rule were te tegrd s ppromted y seod order polyoml We ve, I d d were s seod order polyoml Coose,,,,, evlute d d, s te tree pots o te uto to Solvg te ove tree equtos or ukows, d gves, Numerl Itegrto 87
8 Numerl Metods or Eg [ENGR 9] [Lyes KADEM 7] Te d I d Susttutg vlues o d gves, d Se or Smpso s / rd Rule, te tervl [ ], s roke to segmets, te segmet wdt s Hee te Smpso s / rd rule s gve y d Se te ove orm s / ts ormul, t s lled Smpso s / rd Rule Numerl Itegrto 88
9 Numerl Metods or Eg [ENGR 9] [Lyes KADEM 7] Emple Te dste overed y roket rom 8 t to t s gve y 8 98 l dt t t Use Smpso s / rd Rule to d te ppromte vlue o Fd te true error, t E Fd te solute reltve true error, t Multple Segmet Smpso s / rd Rule Just lke multple-segmet Trpezodl Rule, we sudvde te tervl [ ], to segmets d pply Smpso s / rd Rule over every two segmets Note tt eeds to e eve Dvde tervl [ to equl segmets, ee te segmet wdt ], d d were d d d d d Apply Smpso s / rd Rule over e tervl, d Se,,, te d Numerl Itegrto 89
10 Numerl Metods or Eg [ENGR 9] [Lyes KADEM 7] { } { [ ] } eve odd eve odd d Error Multple Segmet Smpso s / rd Rule Te true error sgle pplto o Smpso s / rd Rule s gve y E t < < ζ ζ, 88 I Multple Segmet Smpso s / rd Rule, te error s te sum o te errors e pplto o Smpso s / rd Rule Te error segmet Smpso s / rd Rule s gve y, 88 E < < ζ ζ 9 ζ, 88 E < < ζ ζ 9 ζ : E < <, 88 ζ ζ 9 ζ Hee, te totl error Multple Segmet Smpso s / rd Rule s t E E 9 ζ Numerl Itegrto 9
11 Numerl Metods or Eg [ENGR 9] [Lyes KADEM 7] Te term were 9 9 ζ ζ ζ s ppromte verge vlue o ζ E t 9, < < Hee Rrdso s Etrpolto Formul or Trpezodl Rule Te true error multple segmet Trpezodl Rule wt segmets or tegrl d s gve y E t ξ were or e, te term ξ s pot somewere te dom [, ] ξ, d e vewed s ppromte verge vlue o us to sy tt te true error, Et e wrtte uder te orm: [,] Ts leds E t α Numerl Itegrto 9
12 Numerl Metods or Eg [ENGR 9] [Lyes KADEM 7] Or C E t were Se, we ve C s ppromte ostt o proportolty E TV t I were TV true vlue I ppromte vlue usg -segmets Te, we wrte, C TV I I te umer o segmets s douled rom to te Trpezodl rule, C TV I Te ove equtos e omed to get: TV I I I GAUSS QUADRATURE RULE Dervto o two-pot Guss Qudrture Rule Te two-pot Guss Qudrture Rule s eteso o te Trpezodl Rule ppromto were te rgumets o te uto re ot predetermed s d, ut s ukows d So te two-pot Guss Qudrture Rule, te tegrl s ppromted s I d Numerl Itegrto 9
13 Numerl Metods or Eg [ENGR 9] [Lyes KADEM 7] Tere re our ukows,, d Tese re oud y ssumg tt te ormul gves et results or tegrtg geerl trd order polyoml, Hee d d Te ormul gves d Equtg te ove equtos gves Se ts equto, te ostts d re rtrry, te oeets o d re equl Ts gves us te our ollowg equtos:,,,,,, d Numerl Itegrto 9
14 Numerl Metods or Eg [ENGR 9] [Lyes KADEM 7] Hee d Hger pot Guss Qudrture Formuls I we wrte te tegrl o te uto uder te ollowg orm: d Ts s lled te tree-pot Guss Qudrture Rule Te oeets, d, d te uto rgumets, d re lulted y ssumg te ormul gves et epressos or tegrtg t order polyoml d Geerl -pot rules would ppromte te tegrl d Argumets d wegg tors or -pot Guss Qudrture Rules Usully oeets d rgumets or -pot Guss Qudrture Rule re tulted But, tey re gve or tegrls o te orm g d g Numerl Itegrto 9
15 Numerl Metods or Eg [ENGR 9] [Lyes KADEM 7] Tle : Wegtg tors d uto rgumets used Guss Qudrture ormuls Wegtg Futo Pots Ftors Argumets Note: te tle s gve or g d tegrls, ow we solve d? Ay tegrl wt lmts o [, ] e overted to tegrl wt lmts [, ] Let Numerl Itegrto 9
16 Numerl Metods or Eg [ENGR 9] [Lyes KADEM 7] mt I, te t I, te t su tt m m Solvg tese two smulteous ler Equtos gves m Hee t d dt Susttutg our vlues o d d to te tegrl gves us d d Emple Use tree-pot Guss Qudrture Rule to ppromte te dste overed y roket rom t 8 to t s gve y 8 l 98t dt t Also, d te solute reltve true error Numerl Itegrto 9
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