Trapezoidal Rule of Integration

Transcription

1 Trpezoidl Rule o Itegrtio Civil Egieerig Mjors Authors: Autr Kw, Chrlie Brker Trsormig Numericl Methods Eductio or STEM Udergrdutes /0/00

2 Trpezoidl Rule o Itegrtio

3 Wht is Itegrtio Itegrtio: The process o mesurig the re uder uctio plotted o grph. y x dx x I x dx Where: x is the itegrd lower limit o itegrtio upper limit o itegrtio x 3

4 Bsis o Trpezoidl Rule Trpezoidl Rule is sed o the Newto-Cotes Formul tht sttes i oe c pproximte the itegrd s th order polyomil I x dx where x x d x x... 0 x x 4

5 Bsis o Trpezoidl Rule The the itegrl o tht uctio is pproximted y the itegrl o tht th order polyomil. x x Trpezoidl Rule ssumes, tht is, the re uder the lier polyomil, x dx 5

6 Derivtio o the Trpezoidl Rule 6

7 Method Derived From Geometry The re uder the curve is trpezoid. The itegrl y x dx x x dx Sum o Are o prllel trpezoid sides height x Figure : Geometric Represettio x 7

8 Exmple The cocetrtio o ezee t criticl loctio is give y e erc [ erc ] c.75 x z where erc x e dz z So i the ove ormul erc e dz z e Sice decys rpidly s, we will pproximte.6560 e z erc 0 dz 5 z Use sigle segmet Trpezoidl rule to id the vlue o erc Fid the true error, E t or prt. c Fid the solute reltive true error, or prt.

9 Solutio I z e z 5 e e

10 Solutio cot I The exct vlue o the ove itegrl cot e oud. We ssume the vlue otied y dptive umericl itegrtio usig Mple s the exct vlue or clcultig the true error d reltive true error. erc z e dz

11 Solutio cot True Vlue Approximte Vlue E t c The solute reltive true error, t, would e t True Error True Vlue %

12 Multiple Segmet Trpezoidl Rule I Exmple, the true error usig sigle segmet trpezoidl rule ws lrge. We c divide the itervl [8,30] ito [8,9] d [9,30] itervls d pply Trpezoidl rule over ech segmet t 000l 9. 8t t 30 9 t dt t dt t dt

13 Multiple Segmet Trpezoidl Rule With m / s Hece: m / s m / s t dt m 3

14 Multiple Segmet Trpezoidl Rule The true error is: E t m The true error ow is reduced rom -807 m to -05 m. Extedig this procedure to divide the itervl ito equl segmets to pply the Trpezoidl rule; the sum o the results otied or ech segmet is the pproximte vlue o the itegrl. 4

15 Multiple Segmet Trpezoidl Rule Divide ito equl segmets s show i Figure 4. The the width o ech segmet is: h The itegrl I is: y x I x dx x Figure 4: Multiple 4 Segmet Trpezoidl Rule 5

16 h h h h h h x dx x dx... x dx x dx 6 Multiple Segmet Trpezoidl Rule The itegrl I c e roke ito h itegrls s: x dx Applyig Trpezoidl rule o ech segmet gives: x dx ih i

17 Exmple The cocetrtio o ezee t criticl loctio is give y 3.73 c.75 erc e erc z where erc x e dz Sice e z decys rpidly s, we will pproximte e z erc 0 dz [ ] x z So i the ove ormul erc e dz 5 z Use two-segmet Trpezoidl rule to id the vlue o erc Fid the true error, E t or prt. c Fid the solute reltive true error, or prt.

18 Solutio The solutio usig -segmet Trpezoidl rule is ih I i h

19 Solutio cot The: I i 5 ih [ ] [ ]

20 Solutio cot The exct vlue o the ove itegrl cot e oud. We ssume the vlue otied y dptive umericl itegrtio usig Mple s the exct vlue or clcultig the true error d reltive true error. erc so the true error is z e dz E t True Vlue Approximte Vlue

21 Solutio cot c The solute reltive true error, t, would e t True Error True Vlue %

22 Solutio cot Tle gives the vlues otied usig multiple segmet Trpezoidl rule or: e z erc 0 dz 5 Vl ue E t % % t Tle : Multiple Segmet Trpezoidl Rule Vlues

23 Exmple 3 Use Multiple Segmet Trpezoidl Rule to id the re uder the curve. x 300x x e rom x 0 to x Usig two segmets, we get h 5 d e e e

24 Solutio ih I i i [ ] [ ] The:

25 Solutio cot So wht is the true vlue o this itegrl? x e x dx Mkig the solute reltive true error: t 00% %

26 Solutio cot Tle : Vlues otied usig Multiple Segmet Trpezoidl Rule or: 0 300x x dx 0 e Approximte Vlue % % % % % % % Et t

27 Error i Multiple Segmet Trpezoidl Rule The true error or sigle segmet Trpezoidl rule is give y: 3 E t " ζ, < ζ < where ζ is some poit i [,] Wht is the error, the i the multiple segmet Trpezoidl rule? It will e simply the sum o the errors rom ech segmet, where the error i ech segmet is tht o the sigle segmet Trpezoidl rule. 7 The error i ech segmet is [ h ] 3 E " ζ, < ζ < 3 h " ζ h

28 Error i Multiple Segmet Trpezoidl Rule Similrly: [ ih i h ] 3 Ei " ζ i, i h < ζ i < 3 h " ζ i ih It the ollows tht: [ { h} ] 3 E " ζ, h < ζ < 3 h " ζ 8

29 Error i Multiple Segmet Trpezoidl Rule Hece the totl error i multiple segmet Trpezoidl rule is E t E i i 3 h i " ζ i 3 i " ζ i The term i " ζ i is pproximte verge vlue o the " x, < x < Hece: E t 3 i " ζ i 9

30 Error i Multiple Segmet Trpezoidl Rule Below is the tle or the itegrl l 9. 8t dt t 30 8 s uctio o the umer o segmets. You c visulize tht s the umer o segmets re douled, the true error gets pproximtely qurtered. Vlue E % % t t

31 Additiol Resources For ll resources o this topic such s digitl udiovisul lectures, primers, textook chpters, multiple-choice tests, worksheets i MATLAB, MATHEMATICA, MthCd d MAPLE, logs, relted physicl prolems, plese visit _rule.html

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