unidade 5 - inegração numérica

6
y = f (x) [a, b] P n (x) f (x) f (x) (x 0 ,f (x 0 )), (x 1 ,f (x 1 )), (x 2 ,f (x 2 )) x 0 = a x n = b f (x) f (x) f (x) t 0 s (t 3/2 - s 3/2 ) 2/3 ds

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Unidade 5 - Inegração Numérica.pdf

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Page 1: Unidade 5 - Inegração Numérica

!"#$%&' (

!"#$%&'() *+,-%./&

!"#$%&% !'(#%)*&(#!"# '(& +'!,-. y = f(x) #( '( /&/. )!"#%0&1. [a, b] 2

)!"#$%&% '( 3.1)!4(). Pn(x) 5'# &3%.6)(# f(x) !. /&/. )!"#%0&1.7

8.( +%#5'9!*)&: & +'!,-. f(x) 2 /&/& !& +.%(& /# "&;#1&: .' <#=&: #( 3&%#<

.%/#!&/.<: "&)< *.(. (x0, f(x0)), (x1, f(x1)), (x2, f(x2)): .!/# x0 = a # xn = b7

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#: *.!<#5'#!"#(#!"#: )!"#$%&% f(x)7

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/# 3%.*#<<&(#!".< /# /&/.<: *.(.B

∫ t

0

s

(t3/2 − s3/2)2/3ds

;C E *.!F#*)/& & <.1',-. &!&1?")*& /. %#<'1"&/. /& )!"#$%&1: (&< <#' *G1*'1. <H

3./# <#% .;")/. 3.% &3%.6)(&,-.I

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Page 2: Unidade 5 - Inegração Numérica

!" #$%&'( )$ *$+,-./0-,$( 1

!" #$%&'( )$ *$+,-./0-,$(

2 34.56- ' ($& 7.,$%&')' 8 '9&-:7;')' 9$<- 9-<7.=;7- 7.,$&9-<')-&! >( ;8,-)-(

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7.,$&9-<'56-!

!"!" #$%&' () *&'+,-.)

@'&' - 7.,$&A'<- [a, b] )')-B ' 34.56- f(x) 8 '9&-:7;')' 9-& 4; 9-<7.=;7- )$

%&'4 7%4'< ' "B 4;' A$? C4$ (6- D-.(7)$&')-( E 9-.,-( )$ f(x) Fx0 $ x1G!

H7%4&' !"I H4.56- )$ 9&7;$7&- %&'4 C4$ 9'&' ' '9&-:7;'56- )$ f(x)!

*-,$B )$ 'D-&)- D-; ' J%4&' !"B C4$ 7.,$%&'& ' 34.56- 7.,$&9-<')' Pn(x) 8 -

;$(;- C4$ D'<D4<'& ' +,'" &) $,"(-.%) 3-&;')'!

∫ b

a

P1(x)dx =h

2[f(x0) + f(x1)] F !"G

-.)$ h = b− aB ($.)- $(,' D-.K$D7)' D-;- ,'/," &) $,"(-.%)!

!"#$%& '#(()%"&!*( + ,-."!.& /!01*#"&

Page 3: Unidade 5 - Inegração Numérica

!" #$%&'( )$ *$+,-./0-,$( 12

!"#$%& '()() !"#$"!% ! &'()*%!" '$+,%&#! -! .$'/01

I =

∫1

0

1

1 + x2dx

21"$/01

h = 1− 0 = 1

f(0) = 1

1+0= 1

f(1) = 1

1+1= 0, 5

3-%-4

I =

∫1

0

1

1 + x2dx =

1

2[1 + 0, 5] = 0, 75

5 6'7-& '8&-9:;')- )' :.,$%&'7 < )$ 0, 75! =-& ($& >; ;<,-)- .>;<&:?-4 $9:(,$4

?-.@-&;$ -A($&6')- .' B%>&' !"4 >; $&&- '((-?:')- '- &$(>7,')-4 - C>'7 8-)$ ($&

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N (>A/:.,$&6'7-( )$ ';87:,>)$ h = b−an4 -.)$ x0 = aJ xN = b $ $; ?')' (>A/:.,$&6'7-

[xj, xj+1]4 j = 0, 1, ..., N − 14 '87:?'& ' %)*%! -1 (%!3,4&1! 5 $&&- < - &$(>7,')- )'

(-;' )'( K&$'( $.,&$ '( ?>&6'( $ '( &$,'(4 ?-;- ;-(,&')- .' B%>&' !L!

M %)*%! -1 (%!3,4&1 *)')%!"&4!-! < )')' 8$7' $C>'EI-N

∫ xN

x0

f(x)dx =h

2{f(x0) + 2[f(x1) + f(x2) + ...f(xN−1)] + f(xN)} O !LP

*-,$ C>$ - 8&:;$:&- $ - Q7,:;- 6'7-& )$ f(x) .I- (I- ;>7,:87:?')-( 8-& 2!

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Page 4: Unidade 5 - Inegração Numérica

!" #$%&'( )$ *$+,-./0-,$( 12

34%5&' !67 #$8&$($.,'9:- %&;<=' )- >?,-)- )$ ,&'8?@4- %$.$&'A4@')-!

!"#$%& '()(* !"#$%& & '()!*+&$ ,#- h = 0.2.

∫1,2

0

excos(x)

B 8'&,4& )$(,'( 4.C-&>'9D$(E 8-)$>-( =-.(,&54& 5>' ,'F$A' =-> -( G'A-&$( )$ f(x)

G'&4'.)- )$ 0 ' 1, 2E G'&4'.)- 8-& 5> 8'((- h7

x H HE6 HEI HEJ HEK " "E6

excos(x) " "E"L2 "E12I "E H1 "E 6 "EIJK "E6H6

M$ '=-&)- =-> ' CN&>5A' !6E ' 4.,$%&'A ? )')' 8-&7

∫1,2

0

excos(x) =0, 2

2[1 + 2(1, 197 + 1, 374 + 1, 503 + 1, 552 + 1, 468) + 1, 202]

∫1,2

0

excos(x) = 1, 639

!"#$%& '#(()%"&!*( + ,-."!.& /!01*#"&

Page 5: Unidade 5 - Inegração Numérica

!" #$%&'( )$ *$+,-./0-,$( 12

!"!# $%&'( ")# *% +,-./01

3'&' - 45,-)- )-( ,&'6578-(9 :-&'4 ;,8<87')-( 2 6-.,-( =-.($=;,8>-( )$ f(x)

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*$(,$ ($.,8)-9 6-)$/($ ;,8<87'& 3 6-.,-( =-.($=;,8>-( 6'&' ($ -B,$& ;4' :;.?@- )$

($%;.)- %&'; 6'&' - =A<=;<- )' 8.,$%&'<! C$(,' :-&4'9 '- <-.%- )- 8.,$&>'<- [xj, xj+2]9

j = 0, 2, ..., 2N − 29 ,$4-(D

∫ x0N

x0

f(x)dx =h

3[f(x0) + 4f(x1) + f(x2)] E !1F

G(,' 5 )$.-48.')' !" # $%& '! ()*+,-.!

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∫ x0N

x0

f(x)dx =h

3{f(x0)+4[f(x1)+f(x3)+ ...]+2[f(x2)+f(x4)...]+f(xN)} E !JF

*-,$ K;$9 )$.,&- )-( ,$&4-( 4;<,86<8=')-( 6-& h/39 =-4 $L=$((@- )$ f(x0) $ f(xN)9

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=-4 M.)8=$( 6'&$( (@- 4;<,86<8=')-( 6-& 2!

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).0!" #1 '- !4!*+1- 56$676

∫1,2

0

excos(x) =h

3{f(x0)+4[f(x1)+f(x3)+f(x5)]+2[f(x2)+f(x4)+f(x5)+f(xN)}

∫1,2

0

excos(x) =0, 2

3[1 + 4(1, 197 + 1, 503 + 1, 468) + 2(1, 374 + 1, 552) + 1, 202]

!"#$%& '#(()%"&!*( + ,-."!.& /!01*#"&

Page 6: Unidade 5 - Inegração Numérica

!"!#$%&'() *'+,'-.#/"'&() !

∫1,2

0

excos(x) = 1, 639

!"#$%$&' ()* !"#"$%&'( ( )*!('( '( !+%,*$"( -.&.+%#"$%'(/ '.!.+)"&. %0 0.-1"&!.0

"&!.-+%"02

%3

∫1,3

1

√xdx/ 4() h = 0, 055

63

∫0,8

0

cos(x)dx/ 4() h = 0, 15

43

∫1,6

1,2

senxdx/ 4() h = 0, 25

'3

∫0,8

0

xexdx/ 4() h = 0, 15

!"#$%$&' ()+ !"#"$%&'( ( )*!('( '. 7"),0(& -.&.+%#"$%'(/ 4%#41#. %0 "&!.-+%"0

'( .8.+494"( :;<;

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