NUmERIen UNTEGRATHON
DIteLNin+oni
Neimlates Suad egalirntpueas1
oyt
Ihe Neustov- Cotes lomulas ase
Cemmo meiLCal
SAtCgclion h r n .
Ihey ane baseel omhe uadey oh rsplauirg a
Csmplicated f lton
tabulateel dala wil
a
appioxim aling fernclion tholis e alo
IntezsolE
Z
Coh
pb
nb
fCa)loe E Y.e)d
(x)=a polynonial ahe fon
)
Qo +ata,X t
Clard frd Open Toren ¢ Naton Cots
radass
uhee tu dala
tkose
a
e
Closeld
fots
he
limilk
he
enol
ef
Points at The begimming andd
oinls
Ittegalion aie knrun
luits
tntegcalm
h
a
e
lhaCpen forms
extendt beyond ha samee efhe dkla.
New law otes fotmulas
anl met
tkak
generalle
Open
Open
as
wbed fou depimil inteacallin.Houeue Moy a
e
intcys
mpiope
uleCies foeyatualb
ha Sotaliin
foolinauy
feeilial e
foot
Open mteqalis
Closeel Soctegatiom
TRAPETOIDAL RUEE
the tsapezetelal ule.
Neulsn
Cstes tbseel
is th fast of lha
integilin foiulas Jt
Coxespomds to te Case toheré the potymaal
in
(is s t dueler
whoa
x)- fla)+ f6)fa)) (a-a
(x-ald
Pta)-to)b--fta)
a
1
b-a) fla)+fl6)
2
which i Called t
apez2ctdal Ale
hapezpiclat
Ofhe
Sinple Applicalion
Kues
Elomepte,
tuape3oiolal
Auk to mumeaicaly
9nbegnaleE
x)= 0.2+2S-200x+675-ox+toxS
azo
om
t b = 0.8
Prnalytion Solzlion
6y
hapezsickal
1.640533
=
kule
t
7= b-a) fla) tf[L)
L (o.8-o) fo)+{o
a)=
L
e
b)- (o.8) =0.232
0.2
(0.8) fo-2+0-232]-
0.1728
2
valut
Kelalrie
Eann
=
E
C
Perce
=
Aetual a l u e
-
ppsouirnatod
e1.6105 33
selaline
esso
=
0.172 8
= 99 5 ,
=
-
Ct
1.467733
Multiple-Aopleilion opejotdal kuders
Cha uaN lo imnpsoe tthe acc
tiapezoiclal Aule
isto olivicde o
acy f
the
integraliom nte
%om a to b ito a ainloe ef Segmenl auol
appy The melkol to eaul Sepmnb. L aRas Of
iondivicdeal Segmenlz cam ten be added to
Yielcl
yiela tie ivrteal fou
te
entie inter/al he
Called muliple oplcalsn-
esculti9 3uats a e
Cmposite ntesalio fstmulas.
lheke
ae
nt1
egually Space base peEmts
do1, 7 - --n). Conseauely ,
tkise ac n
Segmenls ef eual usiokftu,
h
Y
9fa nd b ane deigmaled as
ospeilvey the total intespo Cam
as
o
addn
be
sLpsesenlee
o
Substitulung ha topejotola Aule 9ov ear Sapomen
stegalyields.
2
th6n-)
O c i y tesmrs
-
foreo) t 2f(xi) +ftx.)
T- h
Etonplo
Use ta
to estimate he
tn
ule
tapezoiolal
3egnmei
inteyal aef
f ) = 0.2+25-200x+ 67S%*-900t+4o
sm
hese
hexe
a s 0 to
a e l h O.8-0- o.4
2
2
So
f(o)
1
-
=
b=0.8
0.21
h2 fo)
O.
Co-4)-2-456
flo.8)-0.232
+2 {(0.4)-+/0.8)
2 +2.4SL)+0-232|1
2
2
= 1.0688)
EncpleEvaluate
d
wih ftut Sub intervals
bt tuape3pidal ule,an
gxau 9platt on = O.2S
h
1-
- O.2
w)+2fxt) 4f{x«)|
Cx)
O-O0
0-2
.
O. y
O.6
O -06y
. 216
O8
O.512
1
1
,
O.
008
o.00
2 o.008-++O- 664+O.21
S12]+ 1
13
SO
O.2
xaut Sotatin
Appto
o. 25
Sotine
so=
o.
2
0.25-0. 26
O.as
O.04
4
SMPsON'Kuts
omdhee
i
hee
polyniomicads
V
fos exanple, i u e is
betureen fCa) ad 6 )
an
ae
i
wit
6
extta
l
te poinls.
poênt micltway
be
h i e potnls Ca
uo
fC6)
potnts eqpually Spaced
he
poinls
Can
be
b/u
lonneted
polymonial
e
hiad -oldes
intoal
sesult kon takionz tk
paabday
a
mulas
Cmcle
to
csnnet
wpe
wik a perabota.
Conneteel
Ca) ad
moe
int eyal i t
a
accurate estimal f
higher ol de
obtam a
Cudo
tal
tse
polyneraials
dalled Snpsong
ae
ules
a
a
MPsoN 's 1/3 Kue r
's
Aele Aesalls whon au Seconel
o%der lniteepolatng polymenial Sabstitule i
Snpsn
1/3
b
7=foxyde
f.odu
H la Intezalion a l algebnc manpudatirn,
t k ollawung 9ormula Aesutts
.
7
twher
fo) + 4 fes) +fla))
h= (b-a)/p .
eualo
9E
tonown
as Sinpsom's 4/5ule,
Aa Second Neciton Cotes
closeol Yntrat on
foummuka. Tha Label "113Stes eme
kal i t divideol by 3.
Stpson S 1/z Aule
OUb
em
also be exreMea
f(x)
+4fox)+
fCxo)
6
a
)
T
wicdlk
Aresage heiqht
fut
Singte Hnplicalion
aptan 3 Cule
mple the m2, Yc its a Secend O,dn
Jtegale potymonial
fx)- 0-2 +25%-20o* +675«90040
onn
Q=0 to b=0.8
Kecall -hat -ke exait integral iá
0.8-0 0.4
b-
h
1.64oS33
2
0. 2
=
flo)
=
Co) f
fx) flo.4) 2. 4S
0.232
=
f(0-8)
fCa) - S Cb)
Cay
-
-
I
#
fx)
O-4
o. 2
+a
f(x1) *ft«)}
+4 (2:. 4S6)
+O.232
7-1.3cP467
OR
IE b - a )
f(xo) +4f(xM)+ f(r.)
(0.B ) 0.2
+4 (2. 4c6) +0.232
1.367467
Nuttple Conpeste plicali
Sempsamí Redes
T
)oda
Shbz enlealo
,
n
J f n d a - f ) +rt
d
n
ayoln
Sernpsm Molo
odd
=
nebSas
)1fCx;)+fC)
t4Ct-)+f(an)
3
fCn-) t{(«,.
h
fas)
+
f (1s) *-ft)
2 f ) t f(xa)tflz)+
h Su ef te
odd rdinates
ordiinatee
2
/ Sun of te herncin io
e e n nelintus
naupe
Ealuae
sige
u
Giuem
Sn
vsing Simpsms kule.
h
6
-
Y-flx)
O
1
S
Sun
1
0.62c
0.0G246
1.c0o6
0.12
O 1246
1. 0OG
0-181S O1864
O. 25
O.2474
1.010
O312s .307
1.016S
C0.37s
0.362
O.437SO 42 3
8
0.062S
6.S
O.474
1.0OS
1.O24O
1. 0323
1.042
=
P
8
Sinfpsns
<ale
S
2(t
8)
1 . 2 9 ) +4 (1.oo06+1.0os9
+1
1.0328)+2(1-00RG
Gs+
o1
1.01oS+ 1-O24O
2 . 0 y 2 9 +16.2230 t6.0742
3403)
dx?
0.SOP
Sin
dample
Spsn
7.
Evaluale o
ule
e d
w
Gdowiny
9om=4.
a -1 , b
2
a
7=4
2-
-a
h
e
1-25
3
1-9S
=
2e
Q.182
274828
3.4903
21923
4817
9898
S.S46
3.238 3
42-073890
4
mpsm ( halewe
y
O.25
3.67452
haue,
2
O.25|21oB + 3.694S2 ) +42-9922
+32883) +2(2.4898)
2.4129
cln
3.OS2
+24.32)+S.14S6|
+
(D
Ca
3h
8
n
t
Ae abone
hule
S
9omula
h i c is
mutthle
uos eM
Aule
Called Smpssn s
aptionble ony
3Sumpos
auial
ALMlE
ahen
m
*5/a Buls
tton Sontps