Сборник задач по высшей математике 2 том

E [0,7I"j, 9 E [0; 271").

y

x ______________ 1,',,,,,,,

Puc. 34

B 06m;eM CJIY'IaenepeMeHHbIe r, 9, rp H3MeHHIOTCH B npe,Il;e.JIax r E [0, +00), rp E

cI>0pMYJIa nepexo,Il;a K c<pepH'IeCKHMKOOp,Il;HHaTaM HMeeT BH,Il;

III!(x, y, z) dxdydz = III!(r sin rpcos 9, rsin rpsin 9, rcos 9)r2 sin rpdrd9drp. v v

v = I I I r2 sin rp drd9drp. v

nplllnO>KeHIIIR TpOMHOrO IIIHTerpana

v = Illdxdydz.

v

2. Macca m Te.JIa V C ,Il;aHHoil: nJIOTHOCTbIO p(x, y, z), I',Il;e<PYHKIJ;HH p(x, y, z)

HenpepbIBHa, BbI'IHCJIHeTCHno <p0pMYJIe

m = Illp(x,y,z)dxdydz.

v

3. CTaTH'IeCKHeMOMeHTbI M xy , M xz , Myz Te.JIa V OTHOCHTeJIbHO KOOp,Il;HHaTHbIX nJIOCKocTeil: Oxy, Oxz, Oyz COOTBeTCTBeHHO paBHbI

Mxy = Illzp(x,y,z)dxdydz,	Mxz = Illyp(x,y,z)dxdy'dz,
v	v

Myz = Illxp(x,y,z)dxdydz,

v

I',Il;ep = p(x, y, z) - nJIOTHOCTb Te.JIa V.

170

4. KOOp)l;HHaThl u;eHTpa TH1KeCTH Te.JIa V C Maccoit m Onpe)l;e.JIHIOTCH no <p0P-

Myz	Mxy
Xc = --;;:n;-,	Zc = --;;:n;-,
J1JIH, 60JIee nO)l;p06HO:
Xc = ~ jjjxp(x,y, z) dxdydz,	Yc = ~ jjjyp(x,y,z)dxdydz,
v	v

Zc = ~ jjjzp(x,y, z) dxdydz. v

B 'IaCTHOCTH,eCJIH p == po (Te.JIO O)l;HOPO)l;HO), 3TH <POPMYJIbI ynporu;aIOTCH:

Xc = t jjjxdxdydz,	Yc = t jjjydxdydz,	Zc = t jjjzdxdydz,
v	v	v

r)l;e v - 06'beMTe.JIa V.

5. MOMeHTbI HHepU;HH TeJIa V C nJIOTHOCTbIO p(x, y, z) OTHOCHTe.JIbHO KOOP)l;H-

HaTHblX IIJIOCKocTeit BbI'IHCJIHIOTCHno <p0pMYJIaM

Jxy = jjjz 2p(x,y,Z)dV,	Jyz = jjjx 2p(x,y,z)dv,
v	v

MOMeHTbI HHepU;HH Jx , J y H Jz Te.JIa V OTHOCHTeJIbHO KOOp)l;HHaTHhlx oceit Ox,

Oy H Oz COOTBeTCTBeHHO HaxO)l;HTCH no <p0pMYJIaM

Jx = j j j(y2 + Z2)p(X, y, z) dv,	Jy = jjj(x 2 + z2)p(x,y,z)dv,
v	v

Jz = jjj(x2 + y2)p(x,y,z)dv. v

3.4.1. BbI'IllCJIllTbTPOtiHOti llHTerpa.rr

jjj x 2 yzdxdydz,

V

rAe V - 06JIacTb, OrpaHll'leHHruIIIJIOCKOCTHMll x = 0, y = 0, Z = 0,

x+y+z=l.

Q 06JIaCTb V (pllC. 35) AOCTaTO'fHOIIPOCTO YCTpoeHa, II09TOMY AaHHbIti TPOtiHOti llHTerpa.rr MO:>KHO BbI'fllCJIllTb,llCIIOJIb3YH IIPOll3BOJIbHbIti IIOPHAOK ItIHTerpllpoBaHllH. Tpa.n;IlIJ;IlOHHO IIpoeKTllpyIOT 06JIaCTb V Ha IIJIOCKOCTb Oxy, npllHllMruI IIOJIY'feHHYIO IIpOeKIJ;IlIO B Ka'feCTBe06JIaCTll D (Ha p"C. D -

TpeyroJIbHllK AOB). IIpHMruI, IIapa.rrJIeJIbHruI OCll Oz, IIepeceKaeT rpaHllIJ;y V

B ABYX TO'fKax.AIIIIJIllKaTa IIepBoti TO'fKllpaBHa HYJIIO (TO'fKaBXOAa JIe:>KllT

171

z

Puc. 35

Ha IIJIOCKOCTH Oxy, T. e. z = 0), aIIIIJIHKaTa BTOPOil: TO'fKHpaBHa z = 1- x - y

(IIOCKOJIbKY TO'fKaBbIXO)l,a H3 06JIaCTH V JIe)lCHT Ha IIJIOCKOCTH z = 1- x -					y).
TaKHM o6pa30M,
	l-x-y
J = III x 2yzdxdydz = II x 2ydxdy		I	zdz = ~II x 2y(l-x-y)2dxdy.
v	D	o		D
,nBOil:HOil: HHTerpaJI IIPHBO)l,HM K IIOBTOPHOMY H3BeCTHbIM Y:>Ke cIIoco6oM, 110-
9TOMY )l,eTaJIH OIIYCKaeM.
1	-x
J = ~ I X 2 dx	I y(l- X - y)2 dy =
0'	0
1	I-x
= ~ I	X 2 dx I y(1 + X 2 + y2 -	2x -		2y + 2xy) dy = ... = 2i20'	•
o	0

B'bt"tuc.I!um'b c.I!eiJY'lO'4Ue mpOfl:H'bte U'Hmeepa.l!'bt 6 npilMoyeO.l!'b'H'btx 'K:oopiJU'Hamax:

3.4.2.	III (x + y +z) dxdydz, r)l,e V -					Ky6, OrpaHH'feHHbIil:IIJIOCKOCTHMH
	V		= 1, y = 0,	y = 1, z = 0, z = 1.
	x = 0, x
3.4.3.	II1(1 -		y)xz dxdydz, r)l,e		V	OrpaHH'feHaIIJIOCKOCTHMH x	0,
	V		= 0, x + y + z = 1.
	Y = 0, z
3.4.4.	III x 2y2 z dxdydz,			r)l,e V	-	IIapaJIJIeJIeIIHIIe)l" OrpaHH'feHHbIil:
	V
	IIJIOCKOCTHMH x = 1, x = 3, y = 0, y = 2, z = 2, z = 5.
3.4.5.	ff!		dxdydz	3'r)l,e V		OrpaHH'feHaKOOp)l,HHaTHbIMH IIJIOC-
	I-	(x + y + z + 1)
	KOCTHMH H IIJIOCKOCTblO X + y + z = 1.
3.4.6.	III x dxdxdz, r)l,e V OrpaHH'feHaIJ;HJIHH)l,POM x 2 + y2 = 1 H IIJIOC-

V = ° = 3.

KOCTHMH Z H Z

172

3.4·7.

3.4.8.

III xyz dxdxdz, r)J;e V OrpaHH'IeHaKOOp)J;HHaTHbIMH IIJIOCKOCnl-

v

MH, c¢epoii X 2 + y2 + Z2 = 1 H paCIIOJIO:>KeHa B IIepBOM OKTaHTe.

BbI'IHCJIHTbTpoiiHOii HHTerpaJI

III(x 2 + y2) dxdydz,

V

eCJIH V OrpaHH'IeHaIIJIOCKOCTblO z	2 H IIapa60JIOH)J;OM 2z
= x2 + y2.

z

Puc. 36

a 06JIaCTb V OrpaHH'IeHacBepxy IIJIOCKOCTblO z = 2, a CHH3Y IIapa60JIQ-

x2 +y2

I1)],OM Z	=	2	(pHC.	36). IIepexo)J;HM K IJ;HJIHH)J;pH'IeCKHMKoop)],HHaTaM
x = r cos cp,		y = r sin cp,		z = z. IIpH aTOM IIO)],bIHTerpaJIbHruI ¢yHKIJ;HH IIpe-
o6pa3yeTcH K BH.IJ:Y x 2 + y2 = r2 cos2 cp + r2 sin2 cp = r2. TaKHM o6pa30M,
J = III(x2 + y2) dxdydz = III r3 drdcpdz =						2.".	2	2
						Idcp I r3 dr		Idz =
v				v		0	0	r2
								"2
			= cpl:""Jr3 dr (zl:2) = 2~J(2 - r;) r3 dr =						~6~.•
				0	2	0
lIepexoa.R. 'x;		-qU,//,U'HapU4teC'x;UM 'x;oopaU'HamaM,				6'bt4tUc,//,umb		aa'H'H'bte	mpoii'H'bte
UH,meepa,//,'bt:
3.4.9.	III dxdydz, r)],e V -				OrpaHH'IeHac¢epoii x 2 + y2 + Z2 = 2Rz,

V

KOHYCOM x 2 + y2 = Z2 H co)],ep:>KHT TO'IKY(0,0, R).

173

	2	v'2x-x2		a
3.4.10.	Jdx	J	dy JzJx2 + y2 dz, npe06pa30BaB CHa'IanaK TpofiHo-
	o	0		0
	My HHTerpany.
	2r	v'2rx-x2			V4r2-x2-y2
3.4.11.	Jdx		J	dy	J	dz, npHBeMl CHa'IanaK TpofiHOMY
	o	v'-2rx-x2			o
	HHTerpany.
3.4.12.	BbI'IHCJIHTbnOBTopHblfi HHTerpan

v'f=X2	Vl-x2-y2
J dy	J	(x2 +y2 +z2)dz.
o	o

a ITpe06pa3yeM nOBTopHblfi HHTerpan B TpofiHOfi

JJJ(x 2 + y2 + Z2) dxdydz, v

.n;JI5I 'Iero,HCCJIe.n.y51 npe.n;eJIbI HHTerpHpOBaHH5I B nOBTopHOM HHTerpane, BOCCTaHOBHM 06JIaCTb HHTerpHpOBaHH5I V. OHa OrpaHH'IeHaCHH3Y nJIOCKOCTbIO

z = 0, T.e. nJIOCKOCTblO Oxy, a cBepxy -	nOBepXHOCTblO z = Jl - x2 - y2,
T. e. BepXHefi 'IaCTblOc¢epbI x 2 + y2 + Z2	= 1. 06JIaCTb D JIe:>KHT B nJIOC-

KOCTH Oxy H OrpaHH'IeHaCHH3Y np5lMofi y = 0 (OCblO Ox) H cBepxy JIHHHefi y = VI - X2, T. e. BepXHefi nOJIYOKpY:>KHOCTblO x 2 + y2 = 1. HaKOHeII, npoeKIIH5I D Ha OCb Ox - 3TO OTpe30K [0,1]. ITo Ha3BaHHbIM nOBepXHOCT5IM nocTpoHM 'IepTe:>K06JIaCTH V (pHC. 37), a no COOTBeTCTBYIOIIIHM JIHHH5IM - 06JIaCTb D (pHC. 38).

z

y

1

o	1 x
Puc. 37	Puc. 38

llcXOMl H3 BH.n;a no.n;bIHTerpanbHofi ¢YHKIIHH H BH.n;a 06JIaCTH HHTerpHpoBaHH5I, .n;eJIaeM BbIBO.n; 0 IIeJIeco06pa3HOCTH nepexo.n;a K C¢epH'IeCKHMKOOP-

.n;HHaTaM: x = rsincpcosO, y = rsincpsinO, z = rcoscp. ITPH 3TOM dxdydz = = r2 sin cp drdcpdO, 0 :::; r :::; 1, 0 :::; cp :::; ~, 0 :::; 0 :::; ~. ITo.n;bIHTerpanbHruI

174

q,YHKIJ;IUI

paBHa

X2 + y2 + Z2

=

r2(sin2 CPCOS20 + sin2 cpsin2 0 + COS2 cp)

=

::::: r2(sin2 cp + COS2 cp)

= r2.

TaKllM 06Pa30M,

11"

11"

(X 2 + y2 + Z2) dxdydz =

2"

2"

1

J:=: I I I

ISin cp dO Idcp I

r 4 dr =

v

0

0

0

= - COScpl: ·cpl: .~r51:= ~.

•

Bt>t"l,UC.I/,Umb

noemop1t'bte U1tmeepa.l/,t>t:

1

X

2(X2+y2)

1

l-x

x+y

3.4.13.

I dx Idy

I

dz.

3.4.14.

I

dx

I

dy

I dz.

x2

xy

0

X2+y2

0

0

a

~ JX2+y2

a

a-x

~

3.4.15.

I

dx

dy

dz.

I

I

3.4.16.

I

dx

I

dy

dz.

0

0

3:2 + y2

0

0

I

a

0

~

a

a+Ja2-p2

3.4.17.

Idcp I

pdp

I

dz.

o

0

p

3.4.18.

Bbl'mCJIllTb 06'beM TeJIa OrpaHll'IeHHOrO c<pepoit x 2 + y2 + Z2 = 22

II IIOBepXHOCTblO IIapa60JIOll)J;a 9z = x 2 + y2.

Q TeJIo V

pacIIOJIO)KeHO Ha,rI, IIJIOCKOCTblO Oxy Me)K)J;y

IIOJIyc<pepoit Z2 =

= ';22 -

x2 - y2 II IIapa60JIOll)J;OM Zl

= ~(X2 + y2) (pllC.

39).

z

x

Puc. 39

06'beM V TeJIa V BbI'IllCJIllM IIO <popMYJIe

V = III dxdydz.

V

175

113 CHMMeTpHH TeJIa V OTHOCHTeJIbHO IIJIOCKocTeit Oxy H Oyz 3aKJIIO'"IaeM, '"ITO YA06HO IIepeitTH K U.HJIHHAPH'"IeCKHM KOOPAHHaTaM X = r COS cp, Y ==

= r sin cp, z = Z H BbI'"IHCJIHTb06beM '"IeTBepToit'"IaCTHV, a pe3YJIbTaT YMHa- }KHTb Ha 4.

Z2

V = 4 III rdrdcpdz = 4 II rdrdcp Idz =

f			D	Z1
= 4 II rdrdcp ( ../22 -				r2 - r;) = 4 II ( ../22 -	r2 - r;) rdrdcp.
D				D
,1l,JIH AanbHeitnIHx BbI'"IHCJIeHHitHa,1l;O HaitTH 06JIaCTb D -					IIpoeKu.HIO Ha IIJIOC-
KOCTb Oxy IIpocTpaHcTBeHHoit 06JIaCTH V. ,ll,JIH 9Toro pemHM cHcTeMY
{ X2 + y2 + Z2 = 22,				2
X	2 +y2	= 9z		=>z +9z=22=>z=2,z=-I1.
IIOACTaBJIHH z = 2 (z			= -11 He IIO,1l;XOAHT, T. K. Z ~ 0) BO BTopoe ypaBHe-

HHe CHCTeMbI, HaitAeM, '"ITOc¢epa H IIapa60JIOHA IIepeceKalOTCH B IIJIOCKOCTH

Z = 2 110 OKPJ)KHOCTH x2 + y2 = 18. CJIeAOBaTeJIbHO, 06JIaCTb D 9TO '"Ie-

0:::; r :::; v'iB,°:::; cp :::; ~. TaKHM 06pa30M,
TBepTb Kpyra x2	+ y2 :::; 18 (x ~ 0, y ~ 0), HJIH, B IIOJIHPHbIX KOOPAHHaTax:
v = 411 ()22 - rL r:) rdr<J,p = 4 i<J,p		/(r)22 - rL ~) dr =
D	0	0
	= 4· ~ ( -l../(22 - r 2)3 -	;~) 1:S=

=(22~-8 -9) .211"= 2(22~-35)11". •

3.4.19.z = x2 + y2, Z = 2(x2 + y2), y = x, y = x2.

3.4.20.Z = ../x2 + y2, 3z = x2 + y2.

3.4.21. 4az = 16 -	x2 -	y2, Z = 4 - x - y, x = 0, y = 0, z = 0.
x2 y2	Z2	x2 y2

3.4.22.4" + "9 + T = 1, 4" + "9 = z.

3.4.23.BbI'"IHCJIHTbKOOPAHHaTbI u.eHTpa TIDKeCTH BepxHeit IIOJIOBHHbI ma-

pa pa,1l;Hyca R C u.eHTpOM B Ha'"laneKOOPAHHaT IIpH YCJIOBHH, '"ITO

a	ero IIJIOTHOCTb IIOCTOHHHa H paBHa PO.
	CAeJIaeM CHa'"lanaPHCYHOK (pHC. 40).
	BOCIIOJIb3yeMcH ¢opMYJIaMH
	Xc = i III xdxdydz,	Yc = i III ydxdydz,	Zc = i III zdxdydz,
	v	v	v

176

z

R

-R

R y

Puc. 40

rAe v = 21rf3 - 06'beMnOJIywapa.

lloAbIHTerpaJIbHbIe <pYHKIIlm x H y B qHCJIHTemlX nepBbIX ABYX APo6eit HeqeTHbIe, a 06JIacTb HHTerpHpOBaHH5I V CHMMeTpHqHa OTHOCHTeJIbHO COOTBeTCTBYIOIIIHX nJIOCKocTeit y = 0 H X = o. l103TOMY XC = Yc = o. K 3TOMY IKe BbIBOAY npHXOAHM, HCXOA51 H3 onpeAeJIeHH5I Xc H Yc H CHMMeTpHH TeJIa OTHOCHTeJIbHO KOOPAHHaTHbIX nJIOCKocTeit Oyz H Oxz. OCTaeTC5I BbIqHCJIHTb

III z dxdydz.

V

,II,.n51 3TOro nepeXOAHM K c<pepHqeCKHM KOOPAHHaTaM TaK IKe, KaK B npHMepe 3.4.12. llOJIyqaeM

I lIz dxdydz = I I I r cos 'P . r2 sin 'P drd()d'P =
v	!11"		v
~		~ffin2~d~·1d91r'dr ~ (-C(~42~ln		(01:") -(": I:) ~ _1rf_4.
				1rR4
CJIeAOBaTeJIbHO, Xc = 0, Yc = 0, Zc = illI z dxdydz = 21r~3 = 3:.						•
3.4.24.			v	-3-
			BbIqHCJIHTb KOOPAHHaTbI IIeHTpa TIDKeCTH TeJIa,		OrpaHHqeHHOrO
3.4.25.			napa60JIOHAOM 4x = y2 + Z2 H nJIOCKOCTblO X = 2.
			BbIqHCJIHTb KOOPAHHaTbI IIeHTpa TIDKeCTH TeJIa,		OrpaHHqeHHOrO
			KOHyCOM
3.4.26.			BbIqHCJIHTb KOOPAHHaTbI IIeHTpa TIDKeCTH TeJIa,		OrpaHHqeHHOrO
			napa60JIOHAOM z = x 2 + y2, nJIOCKOCTblO X + y		= 5 H KOOPAH-

HaTHbIMH nJIOCKOCT5IMH.

177

3.4.27. Bbl'IHCJIHTb KOOp.n;~maTbI lI,eHTpa TmKeCTH TeJla, OrpaHHqeHHOro

9JlJlHIICOH.n;OM	x2	y2	Z2
				°(x ~ 0, y ~ 0,
	64 + 49 + 36 = 1
H Koop.n;HHaTHbIMH IIJlOCKOCT5IMH x = 0, y = 0, z =

z ~ 0).

3.4.28.BblqHCJlHTb MOMeHT HHepll,HH IIp5lMOrO KpyroBoro lI,HJlHH.n;pa Pa,Il;H- yca 4 H BbICOTbI 6 OTHOCHTeJlbHO .n;HaMeTpa CeqeHH5I, IIpOXO.D:5lIll,ero Qepe3 lI,eHTp CHMMeTpHH lI,HJlHH.n;pa; IIJlOTHOCTb lI,HJlHH.n;pa IIOCTo51HHa H paBHa Po.

z

x

Puc. 41

Q BBe.n;eM IIp5lMoyrOJlbHYlO CHCTeMY Koop.n;HHaT Oxyz TaK, KaK 0603HaQeHO Ha pHC. 41: OCb lI,HJlHH.n;pa paCIIOJlO:>KeHa Ha OCH Oz, cpe.n;Hee CeQeHHe lI,HJlHH-

.n;pa Jle:>KHT tl IIJlOCKOCTH Oxy. Tor.n;a 3a,Il;aQa CBO.n;HTC5I K BbIQHCJleHHlO Jx -

MOMeHTa HHepll,HH lI,HJlHH.n;pa OTHOCHTeJlbHO OCH Ox. llcIIOJlb3yeM <POPMYJlY

Jx = ffJ(y2+Z2)p o dv, v

r.n;e V - lI,HJlHH.n;p: x 2 + y2 :::;;					16, -3 :::;;		z :::;; 3. IIepeii.n;eM K lI,HJlHH.n;pHQeCKHM
Koop.n;HHaTaM: x = rcos<p, y = rsin<p, z = z, dxdydz								= rdrd<pdz, 0:::;; <p:::;; 211",
°:::;; r :::;; 4, -3	:::;;	z :::;; 3.		OTclO.n;a y2 + Z2			= r2 sin2 <p + Z2		H, CTaJIO 6bITb,
211"		3		4
Jx = Po f	d<p	f	dz f	(r2 sin2 <p + z2)r dr =
o		-3	0
		= Po		211"	3	(4	2	2)	4
				f d<p	f dz	r	sin2 <p + z ;		10=
						4

o-3

=Po J~<p J(64 sin2 <p + 8z2) dz = Po J~<p[32(1 - cos 2<p)z + 8t]

o

-3

0

178

	211"			(	• 2 )	2
= 48po j [4(1 - cos 2cp) + 3] dcp = 48po tcp -					SlllCP	111"
					- 2 -	0 = 672po1l".
	o
nOIIpo6yitTe B3gTb HHTerpaJI B APyroM IIOPMKe:
	211"		4	3		•
	Po j dcp j		r dr	j(r2 sin2 cp + Z2) dz.
	o	0		-3
3.4.29.	Bbl'IHCJIHTbMOMeHT HHepu.HH IIpgMOrO U.HJIHHApa, BblCOTa KOTOPo-
	ro paBHa H H PaAHyc OCHOBaHHg R, OTHOCHTeJIbHO OCH, COAep)Ka-
	meit AHaMeTp OCHOBaHHg U.HJIHHApa.
3.4.30.	HaitTH MOMeHT HHepu.HH KpyrJIOro KOHyca, BblCOTa KOToporo paBHa
	H, a paAHyc OCHOBaHHg R, OTHOCHTeJIbHO AHaMeTpa OCHOBamig.
3.4.31. HaitTH MOMeHTbI HHepu.HH OTHOCHTeJIbHO KOOPAHHaTHbIX IIJIOCKo-
	cTeit TeJIa, OrpaHH'leHHOrOIIJIOCKOCTgMH ~ + ~ + g'x = 0, y = 0,
	z = 0.
3.4.32.	BbI'lHCJIHTbo6beM v H Maccy m TeJIa V, OrpaHH'leHHoroKOHYCOM
	x 2 + y2 = Z2 H IIJIOCKOCTblO Z = 1, eCJIH ero IIJIOTHOCTb p(x, y, z)

IIPOIIOpU.HOHaJIbHa KOOPAHHaTe z C K09<P<PHu.HeHToM IIPOIIOpU.HOHaJIbHOCTH k, k > 0.

z

y

Puc. 42

o Tpe6yeMble BeJIH'lHHbIBbI'lHCJIHMB U.HJIHHAPH'leCKHX°:::; KOOPAHHaTax:°:::; x =

=rcoscp, y = r sin cp, z = z, dxdydz = rdrdcpdz, cp :::; 211", Z :::; 1,

°:::; r :::; 1 (pHC. 42):

179