Trapezoidal Rule of Integration
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1 Trpezoidl Rule o Itegrtio Mjor: All Egieerig Mjors Authors: Autr Kw, Chrlie Brker Trsormig Numericl Methods Eductio or STEM Udergrdutes /0/200
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 ) ) 2 ) 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 2 2 Sum o Are o prllel ) )) ) trpezoid sides ) height ) x) ) ) 2 ) Figure 2: Geometric Represettio x 7
8 Exmple The verticl distce covered y rocket rom t8 to t30 secods is give y: x l 9. 8t dt t ) 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
9 Solutio ) ) I ) ) t ) 2000l 9. 8t t ) 2000l ) ) m / s ) 2000l ) ) m / s 9
10 Solutio cot) ) I ) m ) The exct vlue o the ove itegrl is x l 9. 8t dt t 06 m 0
11 Solutio cot) ) True Vlue Approximte Vlue E t m c) The solute reltive true error, t, would e t % 06
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 ) 2000l 9. 8t t 30 9 t )dt t )dt t )dt 9 8 ) 8 ) 2 9 ) 30 9 ) 9 ) 2 30 ) 2
13 Multiple Segmet Trpezoidl Rule With 8 ) m / s Hece: 30 ) m / s 9 ) m / s t) dt 9 8) 30 9) m 3
14 Multiple Segmet Trpezoidl Rule The true error is: E t 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
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 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
17 Exmple 2 The verticl distce covered y rocket rom to secods is give y: x l 9. 8t dt t ) 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
18 Solutio ) The solutio usig 2-segmet Trpezoidl rule is I ) 2 2 i ih ) ) h
19 Solutio cot) The: I ) ) i ih ) 30 ) 22 4 [ 8 ) 2 9 ) 30 )] 22 4 [ ) ] 266 m 9
20 Solutio cot) ) The exct vlue o the ove itegrl is x l 9. 8t dt t 06 m so the true error is E t True Vlue Approximte Vlue
21 Solutio cot) The solute reltive true error, t, would e t True Error True Vlue % 2
22 Solutio cot) x Tle gives the vlues otied usig multiple segmet Trpezoidl rule or: l 9. 8t dt t % % Vlue E t t Tle : Multiple Segmet Trpezoidl Rule Vlues 22
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 ) ) ) 0 ) 0 5 ) ) e e e 23
24 Solutio The: I ) 2 2 i ih ) ) ) ) 2 2 ) i 0 ) [ 0 ) 2 5 ) 0 )] [ ) 0. 36]
25 Solutio cot) So wht is the true vlue o this itegrl? x e x dx Mkig the solute reltive true error: t 00% % 25
26 Solutio cot) Tle 2: Vlues otied usig Multiple Segmet Trpezoidl Rule or: 0 300x x dx 0 e Approximte Vlue % % % % % % % Et t 26
27 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
28 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
29 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 ) ) 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
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 /topics/trpezoidl _rule.html
32 THE END
Trapezoidal Rule of Integration
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