Сборник задач по высшей математике 2 том

n06epXHOCmH"btM UHmeepaJIOM 6mopoeo

HenpephIBHaJI BeK~

3) MOMeHTbl HHepD;HH OTHOCHTeJIbHO Koop,n;HHaTHblX oceit H Ha'la.JIaKoop,n;HHaT

J", = I l(y2 + z2)pdu,	Jy = II(x 2 + Z2)p du,	Jz = II(X 2 + y2)pdu,
s	s	s

Jo = II(x 2 + y2 + z2)pdu. s

OnPE!AeneHMe M Bbl'iMCneHMenOBepXHOCTHOrO MHTerpana

BTOpOrO pOAa

~IlrIom;a,n;b nOBepXHOCTH S MOlKHO HaitTH no <p0pMYJle

IIdu = ITJI.S.

S

ECJlH p(x, y, z) - nOBepXHOCTHaJI ITJIOTHOCTb MaTepHa.JIbHoit nosepxHoCTH S,

TO ee Macca m HaxO,n;HTCH TaK:

m = IIp(x,y,z)du.

s

IIycTb S - rJla,n;KaJI opHeHTHpOBaHHaH nOBepXHOCTb, Ha KOTOPOit 3a,n;aHa HenpepblBHaJI <PYHKD;HH R(x, y, z), H nYCTb B KalK,n;Oit TO'lKeM nOBepXHOCTH onpe.n;e-

JleHO nOJlOlKHTeJIbHOe HanpaBJIeHHe HOPMa.JIH n(M) (n(M) -

TO~<PYHKD;HH)•

Bbl6epeM TY CTOPOHY S+ nOBepXHOCTH S, ,lI;JlH KOTOPOit yrOJl MelK.n;y e,n;HHH'l- HOit HOPMa.JIbIO n H OCbIO Oz OCTphIit. Tenepb pa306beM nOBepXHOCTb S Ha 'laCTH SI, ... , Sn C ,n;HaMeTpaMH d1 , ••• , dn . 0603Ha'lHM'lepe3LlP1, ... , LlPn ITJIOm;a,n;H cooTBeTCTBYIOm;HX npoeKD;Hit 'lacTeitS1, ... , Sn Ha ITJIOCKOCTb Oxy, a 'lepe3d -

MaKCHMYM 83 'lHCeJId1 , ••• , dn . Bbl6paB B KalK,n;Oit 'lacTHSi npOH3BOJIbHYIO TO'lKY

Mi(Xi, Yi, Zi), COCTaBHM CYMMY

n
L R(Xi, Yi, zi)LlPi,
i=1
KOTOPaJI Ha3hIBaeTCH UHmeepaJlbHoit CYMMoit 6mopoeo	poda ,lI;JlH <PYHKD;HH
R(x, y, z). IIpe,n;eJI HHTerpa.JIbHblX CYMM (OH cym;ecTByeT B	CHJlY HenpephIBHOCTH

R(x, y, z)) npH d -+ 0, KOTOPhIit He 3aBHCHT OT cnoc06a pa36HeHHH nOBepXHOCTH S

Ha 'lacTHH Bbl60pa TO'leKM i , Ha3hIBaeTCH

poda OT <PYHKD;HH R(x, y, z) no nOBepXHOCTH S H 0603Ha'laeTcH

II R(x,y,z)dxdy.

s+

AHa.JIOrH'lHOonpe,n;eJIHIOTCH nOBepXHOCTHble HHTerpa.JIbl BToporo po,n;a

II P(x,y,z)dydz H	II R(x,y,z)dxdz
s+	Q+

220

r,!l;e F

= {P, Q, R} -

E D C Oxy.

Z(X, y), (x, y) E

OT HenpephIBHhIX <PYHKD;Hit P(X, y, Z) H Q(X, y, Z). CYMMa TpeX YK3.3aHHhIX nOBepX-

HOCTHhIX HHTerparrOB BTOpOro po,!l;a Ha3hIBaeTCH o6UfUM nOaepXH.OCmH.1>£M UH.meZpa-

Jl,OM amopozo poiJa H 0603Ha'iaeTCH

II Pdydz+ Qdxdz+ Rdxdy. s+

IIycTb Tenepb nOBepXHOCTb S HMeeT HBHOe npe,!l;CTaBJIeHHe Z =

Tor,!l;a nOBepXHocTHblit HHTerparr BTOPOro pO,!l;a CBO,!l;HTCH K ,!I;BOitHOMY

HHTerparry no 06.rracTH D

II R(x,y,z)dxdy = IIR(x,y,z(x,y»dxdy. s+ D

ECJIH BbI6paHa npOTHBOnOJIOlKHaH CTopOHa S- nOBepXHOCTH S, TO

II R(x, y, z) dxdy = - II R(x, y, z(x, y» dxdy.

s-	D
AHarrOrH'IHOBhI'IHCJIHIOTCHH nOBepXHOCTHhIe HHTerparrhI
II P(x,y,z)r:lydz H	II Q(x,y,z)dxdz.
s+	s+

CBSl3b nOBepXHOCTHblX IIIHTerpanOB nepBOrO III BTOpOrO pOAa

ECJIH a, fJ, 'Y - yrJIbI, 06pa30BaHHhIe COOTBeTCTBeHHO C nOJIOlKHTeJIbHhIMH HarrpaBJIeHHHMH oceit Ox, Oy H Oz, e,!l;HHH'IHoitHopMarrbIO n K BhI6paHHOit CTopOHe

S+ nOBepXHOCTH S, TO CBH3b nOBepXHOCTHhIX HHTerparrOB nepBoro H BTOpOro po,!l;a

BbIpalKaeTCH CJIe,!I;JIOm;HM paseHcTBoM

II Pdydz + Qdxdz+ Rdxdy = II(Pcosa + QcosfJ + Rcos'Y) dO'.

s+ s

IIOCKOJlbKY n = {cos a, cos fJ, cos 'Y}, TO nOBepXHocTHhIit HHTerparr nepBoro pO,!l;a B npasoit 'IaCTH3Toro paseHcTBa MOlKHO 3arrHcaTb KOMnaKTHO B BeKTopHoit <popMe

IIF. ndO', s

BeKTopHoe nOJIe, onpe,!l;eJIeHHOe Ha S.

BeKTopHoe nOJIe F MOlKHO acCOD;HHpOBaTb C nOJIeM CKopoCTeit lKH,!I;KOCTH, npoTeKaIOm;eit 'Iepe3nOBepXHOCTb S. Tor,!l;a HHTerparr

IIF ·ndO' s

MOlKHO HCTOJIKOBhIBaTb MeXaHH'IeCKHKaK o6m;ee KOJIH'IeCTBOlKH,!I;KOCTH, np<>TeKaIOm;ee B e,!l;HHHD;Y BpeMeHH 'Iepe3nOBepXHOCTb S B nOJIOlKHTeJIbHOM HanpaBJIeHHH,

221

T. e. B.n;OJ1b n. IIo9ToMY 9TOT HHTerp3.JI Ha3b1BaeTCH nomotcoM BeKTOPHOro nOJIH F

'1epe3nOBepXHOCTb S.

4.3.1. BbI'IHCJIHTbIIOBepXHocTHblti: HHTerpaJI IIepBOrO pO)1.a

II(1 + ~a+Z)3'

u

r)1.e a - 'IacTbIIJIOCKOCTH x + y + z = 1, 3aKJIIO'IeHHaJIB IIepBoM

OKTaHTe.

y

x

Puc. 49

Q IIoBepxHocTb a MO:lKHO BbIpa3HTb HBHO: Z = 1 - x - y, (x,y) E D, r)1.e

06JIacTb D - TpeyrOJIbHHK, OrpaHH'IeHHblti:IIpHMblMH x = 0, y = 0 H x+y = 1
(pHC. 49). IIPH	aTOM	da = VI		+ (Z~)2 + (Z~)2dxdy			= ../3dxdy. ,naHHbIti:
HHTerpaJI CBO)1.HTCH K )1.BOti:HOMY (IIpH aTOM 3HaMeHaTeJIb IIO)1.bIHTerpaJIbHoti:
<PYHKII,HH paaeH 1 + x + Z = 1 + x + (1 - x - y) = 2 -							y):
00	_ If../3 dxdy		_ ../311dx		1- X	dy	=
If (1 + x + z)3	-	(2 - y)3	-		/	(2 _ y)3
u	t:,.			0	0

Bb4"I,'uc.n.umb noeepXHOCmHb4e UHmeepa.n.b4 nepeoeo poiJa:

4.3.2.	If	da 2'r)1.e a - 'IacTbIIJIOCKOCTH x + y + Z = 1 IIpH
	u	(1 + x + z)
	YCJIOBHH X ~ 0, y ~ 0, Z ~ O.
4.3.3.	II(z + 2x + ~Y) 00, r)1.e a - 'IacTbIIJIOCKOCTH 6x+4y+3z = 11,

u

JIeJKaIII,aJI B I OKTaHTe.

222

4.3.4.BhIqHCJIHTh nOBepXHocTHhIi!: HHTerpaJI nepBOrO pO.II:a

/ j(X2 + y2)d(1,

CT

r.II:e (1 - c<pepa X2 + y2 + Z2 = R2.

a B CHJIY CHMMeTpHH OTHOCHTeJIhHO KOOp.II:HHaTHhIX nJIOCKocTei!: nOBepXHOCTH (1 H nO.II:hIHTerpaJIhHoi!: <PYHKII,HH OrpaHHqHMCH BhIqHCJIeHHeM HHTerpaJIa npH YCJIOBHH x ~ 0, y ~ 0, Z ~ 0 (T. e. B nepBOM OKTaHTe), a pe3YJIhTaT YMHO)KHM Ha 8.

MCnOJIh3YH c<pepHqeCKHe KOOp.II:HHaThI, 3anHrneM napaMeTpHqeCKHe ypaBHeHHH C<pePhI x = RsincpcosO, y = RsincpsinO, z = R cos cp, yqHThIBM, qTO U = 0, v = cpo Tor.II:a

E= (~:f+ (~~)2+ (~~f=

			= (-R sin cp sin 0)2 + (R sin cp cos 0)2 + 0 = R2 sin2 cp,
G = (~~f+ (~~r+ (~~f=
		= (Rcos cp cos 0)2 + (RcoscpsinO)2 + (-Rsincp)2 = R2,
F = ox ox + oy oy + oz OZ =
OU OV		OU OV	OU OV
		= (-Rsin cpsin O)(R cos cp cos 0) + (Rsin cp cos O)(R coscp sin 0) = 0,
J'E-G---F-2 = R2 sin cp,
a 06JIacTh HHTerpHpOBaHHH -				qeTBepTh Kpyra x2 + y2 ~ R2 (0603HaqHM ee
qepe3 B) B napaMeTpHqeCKoi!: <popMe HMeeT BH.II:
R2 sin2 cpcos20 + R2 sin2 cpsin2 0 ~ R2,					0 ~ cp ~~, 0 ~ 0 ~ ~.
OCTaeTcH		BhIpa3HTh	qepe3	napaMeTphI	nO.II:hIHTerpaJIhHYIO <PYHKII.HIO
f(x,y)	= x 2 + y2. Ha c<pepe x2 + y2 + z2 = R2 HMeeM f(x,y) = R2 - Z2 =
=R2 -	R2 cos2 cp = R2(1- cos2 cp). TaKHM 06pa3oM .II:aHHhIi!: HHTerpaJI paBeH
				11"	11"
				"2	"2
8 / R2(1 -		cos2 cp) . R2 sin cp dOdcp = -8R4 /			dcp /(1- cos2 cp) d(cos cp) =

BOO~ CO~ I:

= -8R4. (coscp _ cp) = iR41r. •

4.3.5.BhIqHCJIHTh nOBepXHocTHhIi!: HHTerpaJI

//(x2 + y2)d(1,

CT

r.II:e (1 - c<pepa x 2+ y2 + Z2 = R2 cnoco60M BhI.II:eJIeHHH 0.II:H03HaqHOi!: BeTBH nOBepXHOCTH HHTerpHpOBaHHH (c<pephI). 3TOT npHMep

223

COBna,n,aeT C npHMepOM 4.3.4, HO ero CJI~eT pemHTb HHbIM cna-: C060M.

4.3.6.	IIvx2 + y2 ds,
	S
	2	2	2	= 0 (0 ~ z ~ b).
	r)1.e S - 60KOBruI nOBepXHOCTb KOHyca X 2	+Y2 -	Z2
4.3.7.	abc
	BblqHCJIHTb nJIOnra,n,b TOt!: qacTH napa6oJIoH)1.a BpanreHHH ay =

= x 2 + Z2, KOTOPruI HaxO)1.HTCH B nepBOM OKTaHTe H OrpaHHqeHa nJIOCKOCTbIO y = 2a (a > 0).

z

Puc. 50

a Cnoco6 1.	IIoBepxHocTb, nJIOnra)1.b KOTOPOt!: 6Y)1.eM BblqHCJIHTb, npe)1.CTa-
BHM B BH)1.e y	=	x2	+ Z2		x H	z. CJIe)1.0Ba-
			a	,T. e. KaK <PYHKIJ;HIO nepeMeHHbIX

TeJIbHO, cooTBeTcTBYIOnrruI <popMYJIa)1.J1H nJIOnra)1.H npHMeT BH)1.

S = IIVI + (y~)2 + (y~)2dxdz,

D

r)1.e 06JIacTb D - npOeKIJ;HH nOBepxHocTH Ha nJIOCKOCTb Oxz (pHC. 50). IIa-

pa60JIOH)1. ay		= x2 + z2		nepeceKaeTCH nJIOCKOCTbIO y			= 2a no OKpJ)KHo-
CTH x 2 + Z2		= 2a2 pa,n,Hyca av'2. CJIe)1.0BaTeJIbHO, D					-	qeTBepTb Kpyra
x 2 + z2	~ 2a2 (x		~ 0, z	~ 0). Onpe)1.eJIHM		nO)1.bIHTerpaJIbHYIO <PYHKIJ;HIO.
TJr	,2x,		2z
J'J.MeeMY:r; = a' Yz = (1'
	1 + (Y' )2 + (y')2 = 1 + 4x2 + 4Z2					= a2 + 4(x2		+ Z2)
		:r;	z	a2	a2	a2		·
TaKHM 06pa30M,				~IIva2 + 4(x2 + z2) dxdz.
			S =

D

IIepexo)1.H K nOJIHpHbIM KOOp)1.HHaTaM, nOJIyqaeM

224

= Jay - X2,

Cnoco6 2. IIoBepxHocTh napa60JIOH,IJ;a npe,lJ;CTaBHM B BH,IJ;e z

T. e. KaK <PYHKIJ;HIO nepeMeHHhIX x H y. B aTOM CJIY'laeCOOTBeTCTBYIOIIJ;aH Q:>0PMYJIa HMeeT BH,IJ;

s = jj VI + (Z~)2 + (Z~)2dxdy,

Dl

r,lJ;e Dl - npoeKu;HH nOBepxHocTH Ha nJIOCKOCTh Oxy. 06JIacTh Dl orpaHHqeHa OChlO Oy, napa6oJIoit x = ..;aY H npHMoit y = 2a. Onpe,D;eJIHM nO,lJ;hIHTerpaJIhHYIO <PYHKIJ;HIO. MMeeM

z'

= _

x

x 2 ,

z' =

a

1

+

(

')2

(')2 =

4ay + a2

x

..jay -

y

2Jay-x2 '

Zx

+ Zy

4( ay-x2)·

TaKHM 06pa30M

s = -2 If J 4ay + a

+ 4ay dy

ViiY

dxdy = -2

j2

a

j

i

2

1

J

2

dx

Dl

Jay - x2

0

o

Jay -

x 2

1

j2a

.

X

IViiY

1 IT

(a2 + 4ay )3/2 2a

13 2

= 2"

..ja2 + 4ay dy . arCSIn ..;aY

0

=

2" . '2.

6

10

=

12 ITa. •

o

4.3.8.

HaitTH nJIOIIJ;a,D;h 'lacTHnapa60JIOH,IJ;a 4z

= x 2 + y2,

OTceKaeMoit

4.3.9.

u;HJIHH,IJ;POM y2 = Z H nJIOCKOCThlO z = 3.

BhI'lHCJIHThnOBepxHocTHhIit HHTerpaJI BToporo pO,lJ;a

jjzdxdy + ydxdz + xdydz,

IT

r,lJ;e G' -

BepXHHH CTopOHa nJIOCKOCTH x +y +z = 1, OrpaHH'leHHoit

KOOp,IJ;HHaTHhIMH nJIOCKOCTHMH.

a MHTerpaJI 6Y,IJ;eM BhI'lHCJIHThnOKoMnoHeHTHo, npoeKTHpYH G' Ha pa3HhIe KOOp,lJ;HHaTHhIe nJIOCKOCTH (CM. pHC. 51).

z

y

Puc. 51

8 CoopHH. _ . no "",ewea M8'reMB11IKC.2 J<YPC

225

//zdxdy.

u

BhlPa>KaH flBHO Z qepe3 x H y, CBe.IJ:eM 3TOT HHTerpaJI K .IJ:BOftHOMY HHTerpaJIy

no 6.0AB. IIo.IJ:cTaBJIfIfI z =			1 -	x -	Y B nO.IJ:hIHTerpaJIhHYIO <PYHKII.HIO II
yqHThIBaH, qTO: 0 ~ x ~ 1, 0 ~ y ~ 1 -					x, nOJIyqaeM
					1	I-x
//zdxdy= //(I - x - y )dxdy= /dx						/(I - x - y )dy=
u		DAB			0	0
		jdX[(1-X)y-y;] I:-X = j(I-;X)2 dx=_(I~x)31:=i.
		o			0
Y6e.IJ:HTeCh, qTO OCTaJIhHhle HHTerpaJIhI
		//ydxdz		H	//xdydz
		u			u
npHBO.IJ:flT K TaKOMY :lKe pe3yJIhTaTY.					II03TOMY		HCKOMhlft HHTerpaJI paBeH
3· i =~.								•
B'bt,",uc.I!um'b c.I!eaY'lO'W,ue UHme2pa.l!'bt 6mop020 poaa:
4.3.10.	//yzdydz+xzdxdz+xydxdy, r.IJ:e a -						BHeWHfIfI CTopOHaTeTpa-
	u							0, y = 0,
	3.IJ:pa, OrpaHHqeHHOrO nJIOCKOCTflMH x + y + z = a, x =
	z=O
4.3.11.	/ /	z dxdy, r.IJ:e S -	BHeWHfIfI cTopoHa 3JIJIHnCOH.IJ:a
	S	x 2		y2	Z2
		2+2+2=1.
		abc
4.3.12.	/ /	x 2 dydz+y2 dxdz+z 2 dxdy, r.IJ:e a -					BHeWHfIfI CTopOHa nOBepx-
	u
	HOCTH BepxHeft nOJIyc<pePhI x 2 + y2 + Z2 = a2.
4.3.13.	/ /	x 3 dydz + y3 dxdz + z3 dxdy, r.IJ:e a -					BHeWHfIfI cTopoHa c<pephI
	u
	x 2 + y2 + Z2 = a2.
4.3.14.	/px - y)dxdy + (z - x)dxdz + (y -						z)dydz, r.IJ:e a -	BHeWHfIfI
	u
	cTopoHa KOHHqeCKoft nOBepxHocTH x 2 + y2 = Z2 (0 ~ Z ~ h).
4.3.15.	HaftTH nOTOK BeKTopHoro nOJIfI F(x,y,z) = xi + yj + zk qepe3
	qacTh nOBepXHOCTH 3JIJIHnCOH.IJ:a
		x2		y2	z2
		2	+ b2 + 2			= 1,
		a			c

JIe:lKaJIU'1OB nepBOM OKTaHTe B HanpaBJIeHHH BHewHeft HOPMaJIH.

226

o HCKOMblit IIOTOK paBeH

I IF. ndO' =Jj{xcosa + ycos;3 + zcos')') dO'.

IT

IT

ilOCJIe,u,HHit HHTerpaJI CBO,n:HTCfI K IIOBepXHOCTHbIM HHTerpaJIaM BToporo po,n:a

II xdydz + II ydxdz + II zdxdy,

Dl

D2

D3

r,n:e D1 , D2 , Da - IIpOeKIJ.HH 3JIJIHIICOH,n:a Ha COOTBeTCTBYIOllIHe Koop,n:HHaTHble IIJIOCKOCTH.

PacCMOTpHM,HaIIpHMep,

II zdxdy,

D3

r,n:e z MO}KHO Bblpa3HTb qepe3 x H y H3 ypaBHeHHfI 3JIJIHIICOH,n:a, Da - BHyTpeHHOCTb qeTBepTH 3JIJIHIICa

x 2	y2
2"	+ 2" :::; 1, x ~ 0, y ~ O.
a	b

II zdxdy

D3

paBeH 06'beMYBOCbMOit qacTH 3JIJIHIIcoH,n:a, KOTOPafl, KaK H3BeCTHO, paBHa

~. ~1rabc. AHaJIOrHqHble HHTepIIpeTaIJ.HH MO}KHO ,n:aTb H ,n:pyrHM HHTerpaJIaM,

II03TOMY HCXO,n:Hblit HHTerpaJI I po,n:a, T. e. IIOTOK BeKTopHoro IIOJIfI, paBeH

3 . ~ . ~1rabc = 1r~bc. •

4.3.16. RaitTH IIOTOK BeKTopa F = x 2i-y2j+z2k qepe3 IIoBepxHocTb TeJIa,

	OrpaHHqeHHOrO c<pepoit x 2 + y2 + Z2 = 3R2, IIJIOCKOCTbIO Oxy H
o	O,n:HOIIOJIOCTHbIM rHIIep6oJIOH,n:OM x 2 + y2 - Z2 = R2.
	HMeeM

= Ilx2 cosadO'- Ily 2 cos;3dO' + Ilz2cos')'dO'.

IT IT IT

Ra IIJIOCKOCTH Oxz H Oyz IIoBepxHocTb O' IIpoeKTHpyeTcfI ,n:Ba}K,n:bI, C 06eHX CTOPOH, K TOMY }Ke IIoBepxHocTb O' cHMMeTpHqHa OTHOCHTeJIbHO 3THX IIJIOCKocTeit. IIo3ToMY cooTBeTcTBYIOllIHe HHTerpaJIbI IIOJIyqaIOTCfI HYJIeBbIMH:

IIx 2 cos adO' = IIy2 cos;3dO' = O.

IT IT

A TeIIepb BblqHCJIHM

II Z2 cos ')' dO'.

IT

IIoBepxHocTb O' COCTOHT H3 Tpex qacTeit (CM. pHC. 52):

227

z

y

Puc. 52

a) cerMeHT c<i>epbI Z = J3R2 - x 2 - y2, ,1I,JUI KOToporo cos')'> 0 (BHeIIImlH HopManb 06pa3yeT c 0 Z OCTPbIii yrOJI); npOeKIJ;HH 3Toro cerMeHTa Ha Oxy eCTb Kpyr x 2 + y2 :::; 2R2 (cerMeHT c<i>epbI x 2 + y2 + Z2 = 3R2 nepeceKaeTcH C rHnep6oJIOH.IJ;OM x 2 + y2 - Z2 = R2 no JIHHHH

{X2 + y2 + Z2 x 2 + y2 _ Z2

=

=

3R2	::::} {X2 + y2 = 2R2
R2	Z = R

OKP)')KHOCTbPMHyca V2R);

	6) cerMeHT napa60JIOH.IJ;a npoeKTHpyeTcH Ha Oxy B KOJIbIJ;O R2 :::; x 2+y2 :::;
:::; 2R2, Z = x 2 + y2 - R2 (H3 ypaBHeHHH rHnep6oJIOH.IJ;a);
	B) HaKOHeIJ;, TpeTbH qacTb - 3TO Kpyr x 2 + y2 :::; R2, Ha KOTOPOM Z = O.
IIo3TOMY
IIF . n du = 1/z2 cos')'du =
IT	II	IT	II
=		(3R2-x2 _y2)dxdy-		(X 2+y2_R2)dxdy= 71r2R 4 .
	x2+y2~2R2		R2~x2+y2~2R2
IIpe.n;JIaraeM caMOCTOHTeJIbHO BblqHCJIHTb 3TH HHTerpanbI.					•

4.3.17.HaiiTH nOTOK BeKTopa F = x 2i+y2j + z2 k qepe3 nOBepXHOCTb TeJIa

~J x 2 + y2 :::; Z :::; H B HarrpaBJIeHHH BHeIIIHeii HopManH.

4.3.18.HaiiTH nOTOK BeKTopa F = 2xi - yj qepe3 qacTb nOBepXHOCTH IJ;HJIHH.IJ;pa x 2 + y2 = R2, X ~ 0, y ~ 0, 0 :::; Z :::; H B HanpaBJIeHHH

BHeIIIHeii HopManH.

4.3.19. HaiiTH nOTOK BeKTopa F = x 2i - y2j + z 2 k qepe3 qacTb c<i>epbI x 2 + y2 + Z2 = R2, X ~ 0, y ~ 0, Z ~ 0 B HanpaBJIeHHH BHeIIIHeii HopManH.

228

4.3.20. RaitTH IIOTOK BeKTopa F = xi + yj - 2zk qepe3 IIoBepxHocTh Ky6a

	Ixl :::; a, Iyl :::; a, Izl :::; a B HaIIpaBJIeHHH BHeIIlHeit HopManH.
4.3.21.	RaitTH MacCY IIOJIyc<pePhI x 2 + y2 + Z2 = R2, Z ~ 0 PMHyca R C
	IIoBepxHocTHoit IIJIOTHOCThIO, paBHoit J x 2 + y2.
a MMeeM
		CT
r,IJ:e	Z=JR2-x2_y2, ./I+(z~)2+(z~)2=				R
	V			JR2 - x2 _y2
(IIpoBephTe!). CJIe,rt.oBaTeJIhHO,
		Jx2 +y2		dxdy.
	m=	R	x 2 _
		JR2 -		y2
IIepexoMi K IIOJIHPHhIM KOOp,IJ:HHaTaM B ,IJ:BOitHOM HHTerpane, IIOJIyqaeM
	m = R If r2 drdcp	= R !2~cp	jR	r2	dr = 11"2 R3 .	•
	r~R J R2 - r2	0	0 ..;R2 - r2		2
4.3.22.	RaitTH MacCY IIOBepXHOCTH Ky6a 0:::; x :::; 1, 0 :::; Y :::; 1,0:::; z :::;					1,
	eCJIH IIoBepXHOCTHaH IIJIOTHOCTh B KroK,IJ:Oit ee TOqKe A/(x,y,z)
	paBHa p(x,y,z) = x + y + z.
4.3.23.	RaitTH MacCY IIOBepXHOCTH Ky6a 0:::; x :::; 1, 0:::; Y :::; 1, 0 :::; z :::; 1,
	ecJIH IIOBepXHOCTHaH IIJIOTHOCTh B KroK,IJ:Oit ee TOqKe (x, y, z) paBHa
	p(x, y, z) = xyz.
4.3.24.	OIIpe,IJ:eJIHTh KOOp,IJ:HHaThI u;eHTpa TIDKecTH O,IJ:HOPO,IJ:HOit IIapa60JIH-
	qeCKoit 060JIoqKH az = x 2 + y2 (0:::; Z :::; a).
4.3.25.	RaitTH MOMeHT HHepU;HH qacTH 6oKoBoit IIOBepxHOCTH KOHyca z =

= Jx 2 + y2 (0:::; Z :::; h) OTHOCHTeJIhHO OCH OZ.

4.3.26.BhIqHCJIHTh

CT

r,IJ:e (J' -qacTh IIapa60JIOH,IJ:a z = x 2 + y2, KOTOPaH BhIpe3aHa U;H- JIHH,IJ:POM (x2 + y2) = x 2 _ y2.

AononHMTenbHble 3aAaHMR

4.3.27. RaitTH CTaTHqeCKHe MOMeHThI O,IJ:HOPO,IJ:HOit TpeyrOJIhHoit IIJIacTHHKH x + y + z = a, x ~ 0, y ~ 0, z ~ 0 OTHOCHTeJIhHO KOOp,IJ:HHaTHhIX

IIJIOCKocTeit.

4.3.28. BhIqHCJ1HTh MOMeHT HHepU;HH OTHOCHTeJIhHO Ox c<pepHqeCKoit 060- JIOqKH x2 + y2 + Z2 = R2 (x ~ 0).

4.3.29. BhIqHCJ1HTh MOMeHThI HHepU;HH O,IJ:HOPO,IJ:HOit KOHHqeCKoit o6oJIoqKH

x 2	y2	z2	=0	X	Y	z - b
a2	+ a2 -	b2		(0:::; z :::; b) OTHOCHTeJIhHO IIPHMOit I	='0	=-0-'

229