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by Lale Yurttas, Texas A&M University

Chapter 5 1

Copyright © The McGraw-Hill Companies, Inc. Permission required or reproduction or display.

Chapter 5NUMERICAL

INTEGRATION &DIFFERENTIATION :

NEWTON-COTESINTEGRATION FORMULAS

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2

Numerica Di!ere"tiati#"a"$ I"te%rati#"

• Calculus is the mathematics of change. Becauseengineers must continuously deal with systemsand processes that change, calculus is anessential tool of engineering.

• Standing in the heart of calculus are themathematical concepts of diferentiation and

integration :

x x f x x f

dxdy

x x f x x f

x y

ii x

ii

∆−∆+=

∆−∆+=

∆∆

∆)()(

lim

)()(

0

∫ =b

a

dx x f I )(

Di erentiation Integration

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Chapter 5 3

Figure PT6.2

The integralis e ui!alentto the areaunder the

cur!e.

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Chapter 5 4

Ne t#"-C#te' I"te%rati#"F#rmu a'

• "ost common numerical integrationschemes.

• They are #ased on the strategy ofreplacing a complicated function orta#ulated data with an appro$imating

function that is easy to integrate:

nn

nnn

b

an

b

a

xa xa xaa x f

dx x f dx x f I

++++=

≅=

−−

∫ ∫ 1

110)(

)()(

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Chapter 5 5

Figure 21.1

Si"% e 'trai%hti"e Si"% e para(# a

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Chapter 5 6

Three straightline segments

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T)E TRA*E+OIDAL RULE• The %rst of the &ewton'Cotes closed

integration formulas.• Corresponds to the case where the

polynomial is %rst order:

• The area under this %rst order polynomial isan estimate of the integral of (x) #etween the limits of a and b :

∫ ∫ ≅=b

a

b

a

dx x f dx x f I )()( 1

2

)()()(

b f a f ab I +−=

Trapezoidal rule

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Chapter 5 !

Figure 21.4

(raphicalillustration of thetrape)oidal rule.

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Chapter 5 "

• *hen we employ the integral under astraight line segment to appro$imate theintegral under a cur!e, error may #esu#stantial:

where $ lies somewhere in the inter!alfrom a to b .

3))((121 ab f E t −′′−= ξ

Err#r #, the Trape #i$aRu e

Truncation error

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E.amp e /

+se trape)oidal rule to numericallyintegrate

f $- /.0 1 02$ 3 0//$ 0 1 452$ 6 37//$ 8 1 8//$ 2

from a / to # /.9. The e$act !alueof the integral can #e determinedanalytically to #e .48/266


Chapter 5 1#

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Chapter 5 11

f(x) = 0.2 + 25x – 200x 2 + 675x 3 – 900x 4 + 400x 5

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12

• ;ne way to impro!e the accuracy of thetrape)oidal rule is to di!ide the integrationinter!al from a to # into a num#er of segmentsand apply the method to each segment.

• The areas of indi!idual segments can then #eadded to yield the integral for the entire inter!al.

Su#stituting the trape)oidal rule for each integralyields:

∫ ∫ ∫ −

+++=

==−=

n

n

x

x

x

x

x

x

n

dx x f dx x f dx x f I

xb xan

abh

1

2

1

1

0

)()()(

0

2)()(

2)()(

2)()( 12110 nn x f x f

h x f x f

h x f x f

h I ++++++= −

The Mu tip e-App icati#"Trape #i$a Ru e

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Chapter 5 13

Figure 21.8

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by Lale Yurttas, Texas A&M University Chapter 5 14

• Can #e o#tained #y summing theindi!idual errors for each segment:

Thus, if the num#er of segments n- isdou#led, the truncation error will #euartered.

f nab

E

f ni f

a ′′−−=

′′≅′′

∑2

3

12)(

)(ξ

rr#r # e u p e-App icati#"

Trape #i$a Ru e

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E.amp e 0

+se the two segment trape)oidal ruleto estimate the integral of f $- /.0 1 02$ 3 0//$ 0 1 452$ 6 37//$ 8 1 8//$ 2

from a / to # /.9. The e$act !alueof the integral can #e determinedanalytically to #e .48/266


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SIM*SON1S RULES• "ore accurate estimate of an integral is

o#tained if a high'order polynomial is usedto connect the points.

• The formulas that result from ta<ing theintegrals under such polynomials are calledSimpson’s rules .

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!impson"s #$% rule$parab%la %nne tin' 3 p%ints(

!impson"s %$& rule$ ubi e)uati%n %nne tin' 4 p%ints(

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Simp'#"1' /23 Ru e'

• =esults when a second'orderinterpolating polynomial is used.

by Lale Yurttas, Texas A&M University Chapter 5 1!

[ ]2

)()(4)(3

)())((

))(()(

))(())((

)())((

))((

210

21202

101

2101

200

2010

21

20

2

0

abh x f x f x f

h I

dx x f x x x x

x x x x x f

x x x x x x x x

x f x x x x

x x x x I

xb xa

x

x

−=++≅

−−−−+

−−−−+

−−−−=

==

∫

Simp !"# 1$3 %ule

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1"

*Single segment application of Simpson>s ?6rule has a truncation error of:

*Simpson>s ?6 rule is more accurate thantrape)oidal rule.

ba f ab

E t ξ ξ )(2880

)( )4(5−

−=

Err#r #, the Simp'#"1' /23Ru e

by Lale Yurttas, Texas A&M University Chapter 5

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E.amp e 3

+se the Simpson@s ?6 rule to estimate theintegral of f $- /.0 1 02$ 3 0//$ 0 1 452$ 6 3 7//$ 8 1 8//$ 2

from a / to # /.9. The e$act !alue of theintegral can #e determined analytically to #e

.48/266

by Lale Yurttas, Texas A&M University Chapter 5 2#

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• Aust as the trape)oidal rule, Simpson>s rule can#e impro!ed #y di!iding the integration inter!alinto a num#er of segments of e ual width.

• ields accurate results and considered superiorto trape)oidal rule for most applications.• However , it is limited to cases where !alues are

e uispaced.

• Further , it is limited to situations where thereare an e!en num#er of segments and oddnum#er of points.

The Mu tip e-App icati#"Simp'#"1' /23 Ru e

nab

h −

=

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22

• Total integral:

• Su#stituting the Simpson>s ?6 rule foreach integral yields:

∫ ∫ ∫ −

+++=n

n

x

x

x

x

x

x

dx x f dx x f dx x f I 2

4

2

2

0

)()()(

6)()(4)(2

6)()(4)(

26

)()(4)(2

12

432210

nnn x f x f x f h

x f x f x f h

x f x f x f h I

++++

+++++=

−−


# #

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• Can #e o#tained #y summing theindi!idual errors for each segment.

• The estimated error:

)4(4

5

180)(

f nab

E a−−=

rr#r # e u p e-App icati#"

Simp'#"1' /23 Ru e

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E.amp e 4

+se the multiple'application Simpson@s ?6 rulewith n 8 to estimate the integral of f $- /.0 1 02$ 3 0//$ 0 1 452$ 6 3 7//$ 8 1 8//$ 2


.48/266


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26

• +tili)es when the num#er of segments isodd.

[ ]

3)(

)()(3)(3)(83

)()(

3210

3

abh

x f x f x f x f h

I

dx x f dx x f I b

a

b

a

−=

+++≅

≅= ∫ ∫

Simp'#"1' 32 Ru e

Simp !"# 3$8 %ule


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Err#r #, the Simp'#"1'32 Ru e

• stimated error:

• The 6?9 rule is more accurate thanthe ?6 rule.


)(6480

)( )4(5

ξ f ab

E t −−=

&!re ' ur' e

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E.amp e 5

a- +se the Simpson@s 6?9 rule to estimate theintegral of f $- /.0 1 02$ 3 0//$ 0 1 452$ 6 37//$ 8 1 8//$ 2


.48/266.#- +se it in con unction with Simpson>s ?6 rule to

integrate the same function for %!e segments.

by Lale Yurttas, Texas A&M University Chapter 5 2!

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by Lale Yurttas, Texas A&M University Chapter 5 2"

Simpson>s ?6 and6?9 rules can #e

applied in tandem tohandle multipleapplications with oddnum#ers of inter!als.

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chpter 5 1

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