Trpezoidl Rule o Itegrtio Mjor: All Egieerig Mjors Authors: Autr Kw, Chrlie Brker Trsormig Numericl Methods Eductio or STEM Udergrdutes /0/200
Trpezoidl Rule o Itegrtio
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
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
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 ) ) 2 ) 5
Derivtio o the Trpezoidl Rule 6
Method Derived From Geometry The re uder the curve is trpezoid. The itegrl y x )dx x) x) dx 2 2 Sum o Are o prllel ) )) ) trpezoid sides ) height ) x) ) ) 2 ) Figure 2: Geometric Represettio x 7
Exmple The verticl distce covered y rocket rom t8 to t30 secods is give y: x 30 8 40000 2000l 9. 8t dt 40000 200t ) Use sigle segmet Trpezoidl rule to id the distce covered. ) Fid the true error, E t or prt ). c) Fid the solute reltive true error, or prt ). 8
Solutio ) ) I ) 2 8 30 ) 40000 t ) 2000l 9. 8t 40000 200t 40000 8 ) 2000l 9. 8 8 ) 40000 200 8 ) 77.27 m / s 40000 30 ) 2000l 9. 8 30 ) 40000 200 30 ) 90.67 m / s 9
Solutio cot) ) I 77. 27 90. 67 30 8 ) 2 868 m ) The exct vlue o the ove itegrl is x 30 8 40000 2000l 9. 8t dt 40000 200t 06 m 0
Solutio cot) ) True Vlue Approximte Vlue E t 06 868 807 m c) The solute reltive true error, t, would e 06 868 t 00 7.2959% 06
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 ) 2000l 9. 8t 40000 200t 30 9 t )dt t )dt 8 8 30 9 t )dt 9 8 ) 8 ) 2 9 ) 30 9 ) 9 ) 2 30 ) 2
Multiple Segmet Trpezoidl Rule With 8 ) 77. 27 m / s Hece: 30 ) 90. 67 m / s 9 ) 484. 75 m / s 30 77.27 484.75 484.75 t) dt 9 8) 30 9) 2 2 8 90.67 266 m 3
Multiple Segmet Trpezoidl Rule The true error is: E t 06 266 205 m The true error ow is reduced rom -807 m to -205 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
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 2 4 3 4 x Figure 4: Multiple 4) Segmet Trpezoidl Rule 5
)h )h )h h h h x )dx x )dx... x )dx x )dx 2 2 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 2 2
Exmple 2 The verticl distce covered y rocket rom to secods is give y: x 30 8 40000 2000l 9. 8t dt 40000 200t ) Use two-segmet Trpezoidl rule to id the distce covered. ) Fid the true error, E t or prt ). c) Fid the solute reltive true error, or prt ). 7
Solutio ) The solutio usig 2-segmet Trpezoidl rule is I ) 2 2 i ih ) ) 2 8 30 h 30 8 2 8
Solutio cot) The: I 30 8 2 ) 8 2 2 2 ) i ih ) 30 ) 22 4 [ 8 ) 2 9 ) 30 )] 22 4 [ 77. 27 2 484. 75 ) 90. 67] 266 m 9
Solutio cot) ) The exct vlue o the ove itegrl is x 30 8 40000 2000l 9. 8t dt 40000 200t 06 m so the true error is E t True Vlue Approximte Vlue 06 266 20
Solutio cot) The solute reltive true error, t, would e t True Error True Vlue 00 06 266 06 00.8534% 2
Solutio cot) x Tle gives the vlues otied usig multiple segmet Trpezoidl rule or: 30 8 40000 2000l 9. 8t dt 40000 200t % % Vlue E t t 868-807 7.296 --- 2 266-205.853 5.343 3 53-9.4 0.8265.09 4 3-5.5 0.4655 0.3594 5 094-33.0 0.298 0.669 6 084-22.9 0.2070 0.09082 7 078-6.8 0.52 0.05482 8 074-2.9 0.65 0.03560 Tle : Multiple Segmet Trpezoidl Rule Vlues 22
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 2 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 23
Solutio The: I ) 2 2 i ih ) ) 0 0 2 0 ) 2 0 5 ) 2 2 ) i 0 ) 0 4 0 4 [ 0 ) 2 5 ) 0 )] [ 0 2 0. 039 ) 0. 36] 50.535 24
Solutio cot) So wht is the true vlue o this itegrl? 0 0 300x e x dx 246. 59 Mkig the solute reltive true error: 246. 59 50. 535 t 00% 246. 59 79.506% 25
Solutio cot) Tle 2: Vlues otied usig Multiple Segmet Trpezoidl Rule or: 0 300x x dx 0 e Approximte Vlue 0.68 245.9 99.724% 2 50.535 96.05 79.505% 4 70.6 75.978 30.82% 8 227.04 9.546 7.927% 6 24.70 4.887.982% 32 245.37.222 0.495% 64 246.28 0.305 0.24% Et t 26
Error i Multiple Segmet Trpezoidl Rule The true error or sigle segmet Trpezoidl rule is give y: 3 ) E t " ζ ), < ζ < 2 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. 27 The error i ech segmet is [ h ) ] 3 E " ζ ), < ζ < 2 3 h 2 " ζ ) h
Error i Multiple Segmet Trpezoidl Rule Similrly: [ ih ) i )h )] 3 Ei " ζ i ), i )h < ζ i < 2 3 h " 2 ζ i ) ih It the ollows tht: [ { )h} ] 3 E " ζ ), )h < ζ < 2 3 h 2 " ζ ) 28
Error i Multiple Segmet Trpezoidl Rule Hece the totl error i multiple segmet Trpezoidl rule is E t E i i 3 h 2 i " ζ i ) ) 2 2 3 i " ζ i ) The term i " ζ i ) is pproximte verge vlue o the " x ), < x < Hece: E t 2 ) 2 3 i " ζ i ) 29
Error i Multiple Segmet Trpezoidl Rule Below is the tle or the itegrl 40000 2000l 9. 8t dt 40000 200t 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 2 266-205.854 5.343 4 3-5.5 0.4655 0.3594 8 074-2.9 0.65 0.03560 6 065-3.22 0.0293 0.0040 30
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