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Bafe of multiple Rnteqeal
Tnteqsatfon of^ anct^ îon^ of^ too nosHally calieol mltiole. ntegsatíon. pase o
Muitiple Irtaga
The p&tcular (^) inteataton ot Runetn twe vafable s called doube integs al,and of that
thyee Nasioole is calleddiple mtegzaon.
Some (^) tims (^) ue (^) have (^) to (^) change (^) the (^) NoHiabe (^) to simpltPy (^) the (^) fntegxal (^) while evaluating (^) the (^) mulks -Nay?abl) (^) can (^) be (^) change (^) by chonging dubsiution^ 0%^ by the (^) coosfAate (^) ayetem Lond fndsical) coosclfnate (polan^ apherteal -Maltiple inteosal
Evaluaton of Doukble.
Date
mOAS (^) of (^) lemina (^) mass and (^) walume (^) of (^) cikd
-Doutle (^) inkeqal (^) of Ponctlorpla)
Method- et the segion Cuswe
Page No.
be (^) tuo Acces6e (^) fotegsatan hee (^) ane (^) t cfenent
oNeH the %eqian
R,can
method (^) to (^) evaluae evaluae. aa (^) ouble Rnteaal
ta
U6elu n
evoluanq (^) plane (^) auea
talal and the
i5 boundedded bu the H42la) the and (^) the (^) 1hne laeb
Hence
Method Q
Let the egion (^) Ris (^) bounded ocaly and Iînea
Hence,
NOTE
I all (^) the (^) fouH (^) limRts
Date
Rastbut TÊ floc
dc
the ce
Ae constant,^ hen^ the^ function ean (^) be (^) integsaBed (^) uîth (^) xeLpect (^) to lany yanioble is impleaise and is digtontanvaus
os Rn the b unday af the Seqon of the
Linteqsatan then the (^) chanqt of the ohden wil afect the
0
2
valuate
ANIta?
|Date
4
Lan!
Lan! (^) -tan
tan' 1) - tan
do
DOMS Date
doe
Page No.
L1f a^ egion^ is^
boonded by
Nole fot^
fvaluation of^ Double
s. Zdentiy
Dxow oll^ the^ cHVe^ and^
mayk then
then find^ ehe^ point^ of^
intbioection of^ o)
which even makes the
DOHS
mose thon
Data
nn (^) thethe senionalong
the (^) segion bounded by^ all^ the
6Find 5 the vasat ion of (^) occlos y) along vaxíation a Cos y)
hon one cuty oll the
Pago No
pts
all the cuves
heviz tasiip (^) in (^) the (^) geglo inlenyatian eauie
the CuNe hete its beA miaate.
the Abip and
- (^) We (^) the (^) 1ßmits (^) of (^) ond (^) c (^) louuen mit (^)? aluaus (^) optained (^) som (^) the (^) cuHve (^) whee the (^) iy stats anc alns aptained
is
(^7) The (^) Roalian Bss Rategsated (^) along (^) the (^) stbip (^) (had
and (^) uith (^) gespect (^) to Pos (^) hasipotal (^) stbio
s\asiabion (^) along (^) vetical tsipRs (^) aluays (^) taken uppex foom pot (^) and (^) along (^) the (^) haizontd stip alusoya takentaken 3om left
If (^) yaiation adong (^) the (^) abp ahangqe) (^) withia (^) he
Ex
segian
Evaluate
b
boun ded
alo,t)
3 1imt
A The seqien ef
The ineqsation
Naniable UPisot
Dsawa veybical
artby
to
(
pat
b
stout Psom X_a¥is
ond (^) tesumaat (^) es
dedy
OMS
integ%atRon Rs tne
con be done
ntin
Date
1
Une tifan qle
Page No.
AOPQ
wsth (^) sespect (^) to (^) on
teminate an the ine
ab
fvalval e
ofaf^ the^ chcle^ o
By the e"
herefeve ,two axis
s.puting
2
with
points (a,1) a(,0)
ond
on
ever the^ Pst^ qua^ dyart
CongideH tve^ bemvSe^ in^ ist quandsant)
dydoe
[lae)la- da
doc
dy do
DOMS Date
Page No.
9 The poiat oe Rotese c.ion of ° oc and cbtained
2
poînt of
Hence
1Rot op
os
5 To chang
ntesale st
pasaet to x-ons which atasts Rsom!
Rls of.
he
Hence the
ats
0: Co,^ o)^ and^ o^
Ca)
the
and teHminat.
Eo to
be ws1tien o
aien
Ce O,1 and
(o0)
osdex of
Ptgdyd
O,)
inteqzafon that daaw hosizonta
on the10e
intagsa aft.en Cchangeos
meon
Rsst (^) wst (^) c (^) and (^) th
aThe 3eqion the cobo)
pauallel pauabala
sTo change the
o.leo
hemits of
yThe pauqbala
point (^) of obtaine ntehtan^ oR
1mfts of
to
Can be
bounded
t
3
the
DoMS
Henee (^) o: lo,o)and (^) Qli
Date
pauabola
Hence the qisen înteasal aftethe
Page No.
cdu (^) of (^) înte (^) qoatonthat (^) means
tisst gt
udiauawebaal (^) stnip (^) fis.
4-0x5 whia atawts Rarm theawbic
Rtehminaes (^) on the (^) pana (^) bola yz
the chang
3
DC an
EEvaluate ton 0) t Na2- (^) ddo (^) oVeH (^) the (^) uppen ot he (^) cace (^) s a (^) cos (^) 9) Atasts txom the uppe. LThe segon of fntegzaon thê uppe oP the coscle %ados Nectos oA oheh
cacle
Lont of
3
9
Lo (9.
ds
*79+| Qcos
a2-2)9)
atoso de
lastae)-oa
Date
de
Page No.
Ca2.e)3/
