Trpezoidl Rule o Itegrtio Civil Egieerig Mjors Authors: Autr Kw, Chrlie Brker http://umericlmethods.eg.us.edu Trsormig Numericl Methods Eductio or STEM Udergrdutes /0/00 http://umericlmethods.eg.us.edu
Trpezoidl Rule o Itegrtio http://umericlmethods.eg.us.edu
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 http://umericlmethods.eg.us.edu
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 http://umericlmethods.eg.us.edu
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 http://umericlmethods.eg.us.edu
Derivtio o the Trpezoidl Rule 6 http://umericlmethods.eg.us.edu
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 http://umericlmethods.eg.us.edu
Exmple The cocetrtio o ezee t criticl loctio is give y 3.73 0.6560 e erc 5.758 [ erc ] c.75 x z where erc x e dz z So i the ove ormul erc 0.6560 e dz z e Sice decys rpidly s, we will 0.6560 pproximte.6560 e z erc 0 dz 5 z 0.6560 Use sigle segmet Trpezoidl rule to id the vlue o erc0.6560. Fid the true error, E t or prt. c Fid the solute reltive true error, or prt.
Solutio I 5 0.6560 z e z 5 e 5.3888 0 0.6560 0.6560 e 0.6509
Solutio cot I 0.6560 5.3888 0 0.6509.44 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 0.6560 0.6560 z e dz 5 0.3333
Solutio cot True Vlue Approximte Vlue E t 0.3333.44.099 c The solute reltive true error, t, would e t True Error True Vlue 0.3333 0.3333 350.79% 00.44 00
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. 40000 t 000l 9. 8t 40000 00t 30 9 t dt t dt 8 8 30 9 t dt 9 8 8 9 30 9 9 30 http://umericlmethods.eg.us.edu
Multiple Segmet Trpezoidl Rule With 8 77. 7 m / s Hece: 30 90. 67 m / s 9 484. 75 m / s 30 77.7 484.75 484.75 t dt 9 8 30 9 8 90.67 66 m 3 http://umericlmethods.eg.us.edu
Multiple Segmet Trpezoidl Rule The true error is: E t 06 66 05 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 http://umericlmethods.eg.us.edu
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 4 4 3 4 x Figure 4: Multiple 4 Segmet Trpezoidl Rule 5 http://umericlmethods.eg.us.edu
h h h h h h x dx x dx... x dx x dx http://umericlmethods.eg.us.edu 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
Exmple The cocetrtio o ezee t criticl loctio is give y 3.73 c.75 erc 0.6560 e erc 5.758 z where erc x e dz Sice e z decys rpidly s, we will pproximte 0.6560.6560 e z erc 0 dz [ ] x z So i the ove ormul erc 0.6560 e dz 5 z 0.6560 Use two-segmet Trpezoidl rule to id the vlue o erc0.6560. Fid the true error, E t or prt. c Fid the solute reltive true error, or prt.
Solutio The solutio usig -segmet Trpezoidl rule is ih I i 5 0.6560.7 5 0.6560 h
Solutio cot The: I 0.6560 5 4.344 4 4.344 4 0.70695 i 5 ih 0.6560 [ 5.88 0.6560 ] [.3888 0 0.6509] 0.0003367
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 0.6560 0.6560 z e dz 0. 3333 5 E t True Vlue Approximte Vlue 0.3333 0.3936 0.70695
Solutio cot c The solute reltive true error, t, would e t True Error True Vlue 0.3333 0.3333 5.63% 00 0.70695 00
Solutio cot Tle gives the vlues otied usig multiple segmet Trpezoidl rule or: 0.6560.6560 e z erc 0 dz 5 Vl ue E t % %.44.099 350.79 --- 0.70695 0.3936 5.63 99.793 3 0.488 0.7479 55.787 44.89 4 0.4057 0.09379 9.483 0.34 5 0.3708 0.056957 8.78 9.566 6 0.35 0.03879.380 5.59 7 0.345 0.088 8.9946 3.063 8 0.33475 0.046 6.8383.083 t Tle : Multiple Segmet Trpezoidl Rule Vlues
Exmple 3 Use Multiple Segmet Trpezoidl Rule to id the re uder the curve. x 300x x e rom x 0 to x 0 0 0 Usig two segmets, we get h 5 d 300 0 300 5 300 0 0 0 5 0. 039 0 0. 36 0 5 0 e e e
Solutio ih I i i 0 5 0 0 0 0 [ ] 0 5 0 4 0 [ ] 36 0 039 0 0 4 0.. 50.535 The:
Solutio cot So wht is the true vlue o this itegrl? 0 0 300x e x dx 46. 59 Mkig the solute reltive true error: 46. 59 50. 535 t 00% 46. 59 79.506%
Solutio cot Tle : Vlues otied usig Multiple Segmet Trpezoidl Rule or: 0 300x x dx 0 e Approximte Vlue 0.68 45.9 99.74% 50.535 96.05 79.505% 4 70.6 75.978 30.8% 8 7.04 9.546 7.97% 6 4.70 4.887.98% 3 45.37. 0.495% 64 46.8 0.305 0.4% Et t
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 http://umericlmethods.eg.us.edu
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 http://umericlmethods.eg.us.edu
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 http://umericlmethods.eg.us.edu
Error i Multiple Segmet Trpezoidl Rule Below is the tle or the itegrl 40000 000l 9. 8t dt 40000 00t 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 66-05.854 5.343 4 3-5.5 0.4655 0.3594 8 074 -.9 0.65 0.03560 6 065-3. 0.093 0.0040 30 http://umericlmethods.eg.us.edu
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