Simpson s 1/3 rd Rule of Integration

Transcription

1 Simpso s 1/3 rd Rule o Itegrtio Mjor: All Egieerig Mjors Authors: Autr Kw, Chrlie Brker Trsormig Numericl Methods Eductio or STEM Udergrdutes 1/10/010 1

2 Simpso s 1/3 rd Rule o Itegrtio

3 Wht is Itegrtio? Itegrtio The process o mesurig the re uder curve. y ( )d () I ( )d Where: () is the itegrd lower limit o itegrtio upper limit o itegrtio 3

4 Simpso s 1/3 rd Rule 4

5 Bsis o Simpso s 1/3 rd Rule Trpezoidl rule ws sed o pproimtig the itegrd y irst order polyomil, d the itegrtig the polyomil i the itervl o itegrtio. Simpso s 1/3rd rule is etesio o Trpezoidl rule where the itegrd is pproimted y secod order polyomil. Hece I ( )d ( )d Where ( ) is secod order polyomil. ( ) 0 1 5

6 Bsis o Simpso s 1/3 rd Rule Choose (, ( )),,, d (, ( )) s the three poits o the uctio to evlute 0, 1 d. ( ) ( ) ( ) ( ) 0 1 6

7 Bsis o Simpso s 1/3 rd Rule Solvig the previous equtios or 0, 1 d give ( ) ( ) ( ) 4 4 ( ) 3 ( ) ( ) 3 ( ) ( ) ( ) 4 ( )

8 8 Bsis o Simpso s 1/3 rd Rule The )d ( I ( ) d ) (

9 Bsis o Simpso s 1/3 rd Rule Sustitutig vlues o 0, 1, give ( )d 6 ( ) 4 ( ) Sice or Simpso s 1/3rd Rule, the itervl [, ] is roke ito segmets, the segmet width h 9

10 Bsis o Simpso s 1/3 rd Rule Hece ( )d h 3 ( ) 4 ( ) Becuse the ove orm hs 1/3 i its ormul, it is clled Simpso s 1/3rd Rule. 10

11 Emple 1 The distce covered y rocket rom t8 to t30 is give y l 9. 8t dt t ) Use Simpso s 1/3rd Rule to id the pproimte vlue o ) Fid the true error, Et c) Fid the solute reltive true error, t 11

12 Solutio ) 30 8 ( t) dt ( ) 4 ( ) [ ( 8 ) 4 ( 19 ) ( 30 )] [ ( ) ] m

13 Solutio (cot) ) The ect vlue o the ove itegrl is l 9. 8t dt t m True Error E t m 13

14 Solutio (cot) )c) Asolute reltive true error, t 100% % 14

15 Multiple Segmet Simpso s 1/3rd Rule 15

16 Multiple Segmet Simpso s 1/3 rd Rule Just like i multiple segmet Trpezoidl Rule, oe c sudivide the itervl [, ] ito segmets d pply Simpso s 1/3rd Rule repetedly over every two segmets. Note tht eeds to e eve. Divide itervl [, ] ito equl segmets, hece the segmet width h ( )d 0 ( )d where 0 16

17 Multiple Segmet Simpso s 1/3 rd Rule () ( )d 4 ( )d ( )d ( )d... ( )d Apply Simpso s 1/3rd Rule over ech itervl, ( )d ( 0 ( 0 ) 4 ( 1 ) ( ) 6 ) ( ) ( 3 ) ( ) ( 4 )

18 Multiple Segmet Simpso s 1/3 rd Rule ( 4 ) ( 3 ) (... ( 4 ) 6 4 )... ( ) 4 ( 1 ) ) 6 ( ( ) Sice h i, 4,..., i i 18

19 Multiple Segmet Simpso s 1/3 rd Rule The ( )d ( 0 ) 4 ( 1 ) h 6 ( )... ( ) 4 ( 3 ) ( ) h ( 4 ) 4 ( 3 ) h 6 ( ) ( ) 4 ( ) h 6 1 ( )... 19

20 Multiple Segmet Simpso s 1/3 rd Rule 0 h ( )d ( 0 ) 4{ ( 1 ) ( 3 )... ( 1 )} 3 [...] { ( ) ( )... ( )} ( )}]... 4 h 1 ( ) 4 ( ) ( ) ( ) i i i i i i odd eve 1 ( 0 ) 4 ( ) ( ) ( ) 3 1 i i i i i i odd eve

21 Emple Use 4-segmet Simpso s 1/3rd Rule to pproimte the distce covered y rocket rom t 8 to t30 s give y l 9.8t dt t 1 ) Use our segmet Simpso s 1/3rd Rule to id the pproimte vlue o. ) Fid the true error, E t or prt (). c) Fid the solute reltive true error, or prt ().

22 Solutio ) Usig segmet Simpso s 1/3rd Rule, 30 8 h So ( t 0 ) (8) ( t 1 ) (8 5.5) (13.5) ( t ) ( ) (19) ( t 3 ) (19 5.5) (4.5) ( t 4 ) (30)

23 Solutio (cot.) 1 ( t0 ) 4 ( t ) ( ) ( ) 3 1 i i i i i i t t odd eve (8) 4 ( t ) ( ) 3(4) 1 i i i i i i t odd eve 1 (30) [ (8) 4 ( t ) 4 ( t ) ( t ) (30)] 1 3 3

24 Solutio (cot.) cot [ (8) 4 (13.5) 4 (4.5) (19) (30)] 11 6 [ (30.469) 4( ) ( ) ] m 4

25 Solutio (cot.) ) I this cse, the true error is E t m c) The solute reltive true error t % 0.007% 5

26 Solutio (cot.) Tle 1: Vlues o Simpso s 1/3rd Rule or Emple with multiple segmets Approimte Vlue E t Є t % 0.007% % % % 6

27 Error i the Multiple Segmet Simpso s 1/3 rd Rule The true error i sigle pplictio o Simpso s 1/3rd Rule is give s 5 ( ) (4) E t ( ζ), < ζ < 880 I Multiple Segmet Simpso s 1/3rd Rule, the error is the sum o the errors i ech pplictio o Simpso s 1/3rd Rule. The error i segmet Simpso s 1/3rd Rule is give y 7 E E 1 5 h ) 5 ( 0 ) ( 4 ) ( 4 ( ζ1 ) ( ζ1 ), ( 4 ) ( 4 ) h ( 4 ) ( ζ ) ( ζ ), < 0 < ζ1 < < ζ 4

28 E i ( Error i the Multiple Segmet Simpso s 1/3 rd Rule i 880 ) 5 ( i 1 ) ( 4 ) ( ζ i 5 ) h 90 ( 4 ) ( ζ 5 ( 4 ) 5 ( 4 ) E ζ h ( ) , ζ i ), ( i 1 ) < ζi < i 4 < ζ < 1 8 ( E 880 ) 5 4 ζ h ζ 5 (4), < ζ < 90

29 Error i the Multiple Segmet Simpso s 1/3 rd Rule Hece, the totl error i Multiple Segmet Simpso s 1/3rd Rule is E t E i i 1 5 h 90 i 1 (4) ( ζ i ) ( ) i 1 (4) ( ζ i ) (4) 5 ( ζi ) ( ) i

30 Error i the Multiple Segmet Simpso s 1/3 rd Rule The term i 1 (4) ( ζ i ) is pproimte verge vlue o (4) ( ), < < Hece E t ( ) (4) 30 where (4) i 1 (4) ( ζ i )

31 Additiol Resources For ll resources o this topic such s digitl udiovisul lectures, primers, tetook chpters, multiple-choice tests, worksheets i MATLAB, MATHEMATICA, MthCd d MAPLE, logs, relted physicl prolems, plese visit /topics/simpsos_ 13rd_rule.html

32 THE END