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Санкт-Петербургский государственный университет телекоммуникаций им. проф. М.А. Бонч-Бруевича

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[НЕСОРТИРОВАННОЕ]

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Сборник задач по высшей математике 2 том

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Puc. 28

CJIe,n:OBaTeJIbHO,

	2	v'Y	2
V= jj(2-y)dXdy=2 j(2-y)dy jdx=2 j(2-y)y'ydy=
D	0	0	0

= 2(2. ~y~ _~y~) 1:= 3;~.•

B'bt"l,uc.n,um'b o6t>eM'bt me.n" oepa'ltU"l,e'lt'lt'btX aa'lt'lt'bt.Mu noeepX'ltOCm.H.MU:

3.3.24.z = 0, Z = 3 - X2 _ y2.

3.3.25.x = 0, y = 0, Z = 0, y = 4, Z + X2 + y2 = l.

3.3.26.x = 0, y = 0, Z = 0, ~ + ~ + ~ = l.

3.3.27.x = 0, y = 0, Z = 0, x = y2 + z2, y + Z = l.

3.3.28.az = y2, X2 + y2 = r2, Z = 0.

3.3.29.Z = X2 + y2, Y = X2, Y = 1, Z = 0.

3.3.30.x + y + Z = a, 3x + y = a, 3x + 2y = 2a, y = 0, Z = 0.

3.3.31.		X2	Z2	b
3.3.31.		2"	+ "2	= 1, y = aX, y = 0, Z = 0.
		a	C
3.3.32.		BbI'fHCJIHTbnJIOrn;a,n:b nOBepxHocTH c¢epbI X2 + y2 + Z2 = R2.
a	C¢epa cHMMeTpH'IHaOTHOCHTeJIbHO Koop,n:HHaTHblx nJIOCKocTeil:, n09TO-
My	OrpaHH'IHMC5I BbI'IHCJIeHHeM nJIOrn;a,n:H nOBepXHOCTH TOil: ee 'faCTH, 'ITO

pacnOJIO)KeHa B nepBOM OKTaHTe, a pe3YJIbTaT YMHO)KHM Ha 8. 3anHIIIeM no-
BepXHOCTb		BepxHeil: nOJIyc¢epbI					51BHO,	T. e. B	BH,n:e	Z =	JR2 - X2 - y2, H
BOCnOJIb3yeMc5I cooTBeTcTByrorn;eil: ¢OpMyJIOil:. MMeeM:
		z'	-	-		X	y2 '	z' - -		y
		x	-		JR2 - X2 -		y2 '	Y -	JR2 - X2 _ y2 '
. /1 + Z'2 + Z'2 =					1 +	2		+	2	=	R
. /1 + Z'2 + Z'2 =					1 +	X		+	Y	=	.
V	x	Y		V		R2 _ X2 _ y2		R2 -	X2 -	y2	JR2 _ X2 _ y2

160

Depexo,IT,H K IIOJUlpHbIM KOOp,nHHaTaM x = r cos t.p, y = r sin t.p, Hafi,neM HCKD-

MYiO IIJIOm.a.n.b (3aMeTHM, qTO 3,neCb MbI HMeeM ,neJIO co CXO,IT,Hm.HMCSI Heco6-

C'fBeHHbIMHHTerpaJIOM)					~ R
5 =8 /' r		R	. r drdt.p = 8R fdt.p!			r dr	r2	=
DJ J R2 - r2				0	0	J R2 -	r2	R
						1!:		R
					= 8R· t.pl:		(-JR2 - r2)lo = 47rR2 . •
3.3.33.	BbIqHCJIHTb IIJIOm.a.n.b S qacTH IIoBepxHocTH IIapa60JIOH,na z = xy,
	IIpHHa)J;JIe:>Kam.efi IJ;HJIHH,npy x2 + y2 ~ R2.
o IIoCKOJIbKY z~			= y, z~ = x, Jl		+ Z~2 + Z~2 = vlri-+-X"":;:2-+-y=2, TO, IIepeXO,IT,H
K IIOJISlPHbIM Koop,nHHaTaM, HMeeM:
5 =	ff	Jl +x2 +y2dxdy =			ffr~drdt.p =
x2+y2~R2					r~R
			2~	R				3
			= fdt.p	fJl + r2 . ~d(1 + r2) = 2;[(1 + R2)"i -1]. •

3.3.34.BbIqHCJIHTb IIJIOm.a.n.b qaCTH IIoBepxHocTH IJ;HJIHH,npa x2+ y2 = R2,

3aKJIlOqeHHOfi Me:>K)J;y IIJIOCKOCTSlMH z = 0 H Z = px, p > O.

	z
	x
	x
Puc. 29	Puc. 30

Q IIoBepxHocTb IJ;HJIHH,npa He MO:>KeT 6bITb 3aIIHcaHa SlBHOfi <popMYJIofi z =

== z(x, y), II09TOMY <popMYJIa

S = ffJl + Z~2 + Z~2dxdy

ReIIpHMeHHMa. BbIpa3HM B TaKOM CJIyqae IIOBepXHOCTb IJ;HJIHH,npa (pHC. 29)

}fBRO B BH,ne y = ±J R2 - x2 H BOCIIOJIb3yeMcSI <popMYJIofi

S = ffJl + y~2 + y? dxdz,

r)J,e D - 06JIacTb, OrpaHHqeHHaSl IIpSlMbIMH z = px, z = 0, x = R (pHC. 30) B nJIOCKOCTH Oxz. lIMeSl B BH)J;y 3HaK ± IIepe,n pa,nHKaJIOM, BbIqHCJIHM IIJIOm.a.n.b

6 C60pHHKlII,lUI" no B..eweA MaTeMBTHKe. 2 K)'P"

161

IIOJIOBUHbI

IIOBepXHOCTU, T. e. OIIUCbIBaeMOii ypaBHeHUeM Z =

J R2 - X2, a

pe3YJIbTaT y,LI.BOUM. IIMeeM

I _

. /

'2 _ . /

x 2

Yx = -

JR2 _ x 2 ' yz -

VI + Yx + yz -

+ R2 _ x2

CJIe,LI.OBaTeJIbHO,

S = 2 If

dxdz = 2R !

Idz =

J R2 -

x 2

J R2 -

x 2

= 2pR!

dx = -2pR(-.jR2 - x2)1

= 2pR2 . •

o JR2_X2

3.3.35.

HaiiTu IIJIOm;a,LI.b qaCTU IIoBepxHocTU Z2 + (x cos a + y sin a) = r2

cO,LI.ep)Kam;eiiCH B IIepBOM OKTaHTe.

3.3.36.

HaiiTu IIJIOm;a,LI.b qaCTU IIJIOCKOCTU

~ + ~ + ~ = 1, 3aKJIIOqeHHOii

Me)K,LI.y KOOp,LI.UHaTHbIMu IIJIOCKOCTHMU.

3.3.37.

HaiiTu IIJIOm;a,LI.b qaCTU IIoBepxHocTU IIapa6oJIou,LI.a y2 + Z2 = 4ax,

OTceKaeMoii U;UJIUH,LI.POM y2 = ax U IIJIOCKOCTbIO x = 3a.

3.3.38.

HaiiTu IIJIOm;a,LI.b qaCTU IIoBepxHocTU IIapa60JIOu,LI.a 2z = x 2 + y2,

«BbIpe3aHH0r0»

U;UJIUH,LI.POM (x 2 + y2)2 = x 2 _ y2.

3.3.39.

BbIqUCJIUTb IIJIOm;a,LI.b TOt!: qacTU KOHyca x 2 + y2 -

Z2 = 0, K0-

TOpM JIe)KUT Ha,LI. IIJIOCKOCTbIO Z

= 0 U

OTCeqeHa

IIJIOCKOCTbIO

z=v'2(~+I).

3.3.40.

BbIqUCJIUTb IIJIOm;a,LI.b qacTU IIOBepXHOCTU rUIIep6oJIuqeCKOrO IIa-

pa6oJIou,LI.a 2z = x 2 -

y2, «BbIpe3aHHot!:»

IIJIOCKOCTHMU x - Y = ± 1,

x +y = ±1.

3.3.41.

OIIpe,LI.eJIUTb MacCY KPYrJIOt!: IIJIaCTUHbI pa,LI.uyca R C u;eHTpoM B Ha-

"lane KOOp,LI.UHaT, eCJIU IIOBepXHOCTHaH IIJIOTHOCTb MaTepuana IIJIa-

CTUHbI B TOqKe M(x,y) paBHa p(x,y) = k.jx2 +y2, r,LI.e k > 0 -

<pUKcupoBaHHoe qUCJIO.

a TIepexo)1.H OT ,LI.eKapTOBbIX KOOp,LI.UHaT K IIOJIHPHbIM, UMeeM

m = IIp(x,y)dxdy =

k.jx 2 + y2 dxdy =

x 2+y2:::;R2

= 4k Id<p I r2 dr = 27r;R. •

3.3.42.

HaiiTu MacCY KpyrJIoii IIJIacTUHbI D (x 2+y2 ~ 1) C IIoBepxHocTHoit

IIJIOTHOCTbIO p(x, y) = 3 -

x -

a IIMeeM:

v'l=?

II (3-x-y)dxdy= Idx

(3-x-y)dy=

x2+y2:::;1

-1

-v'l=?

162

1	2 ~	1
= 1[(3-X)Y-Y2Lv~~X2dx= 12(3-x)~dx=
-1		-1
	1	1
	= 16~dx-2 Ix~dx.
	-1	-1

nOCJIe,LI.HHii HHTerpaJI paBeH HYJIIO, KaK HHTerpaJI OT He'ieTHOii<PYHKIJ;HH no CI1MMeTPH'iHOMYOTHOCHTeJIbHO Ha'iaJIYKOOp,LI.HHaT OTpe3KY. n09TOMY, ,LI.eJIruI nO,LI.CTaHOBKY x = sin t, nOJIY'iHM

		11"			11"
1		2"			2"
m = 16~dx = 6		I	V1-sin2 tcostdt = 6		I cos2 tdt =
-1		~			11"
		2			2"
					11"
					2"
				= 3	1(1 + cos 2t) dt = 311".	•
					11"
					2"
3.3.43. HaiiTH	MOMeHTbI		HHepIJ;HH	KBa,LI.paTHOii	nJIaCTHHbI 0 ~ x ~	a,
o ~ y	~ a	OTHOCHTeJIbHO		oceii KOOp,LI.HHaT H Ha'iaJIaKOOP,LI.H-

HaT, eCJIH nJIOTHOCTb nJIaCTHHbI npOnOpIJ;HOHaJIbHa Op,LI.HHaTe TO'i-

KH nJIaCTHHbI C K09<P<PHIJ;HeHTOM k.

Q BbI'iHCJIeHHlInpOH3BO,LI.HM no COOTBeTCTBYIOIII.HM <popMYJIaM 9Toro napa-

rpa<pa Y'iHTbIBruI,'iTOp(x, y) = xy:
			a	a
1) Jx	=	II	ky· y2 dxdy = k I dx I y3 dy = k~5.
		O~x~a	0	0
		O~y~a
2) Jy	=	II	ky· x 2 dxdy = k I ax 2 dx I ay dy = k~5 .
		O~z~a	0	0
		O:S;;lI~a

•

Hai1mu Macey n,l/,aCmUH'bt D c n06epxHocmHoi1 n,l/,OmHOCmb10 p(x, y):

3.3.44.	D: 0	~ x ~ 1,	0	~ y ~ 2; p(x, y) = xy.
3.3.45.	D: 0	~ x ~ 1,	0	2
3.3.45.	D: 0	~ x ~ 1,	0	~ y ~ 1; p(x,y) =~.
				1+y

3.3.46.D: 0 ~ x ~ 1, 0 ~ y ~ 2; p(x,y) = x 2yeXY .

3.3.47.D OrpaHH'ieHaKpHBbIMH x 2+y2 = ax, x 2+y2 = 2ax, y = 0 (y > 0);

p(x,y) = x 2 + y2.

3.3.48.D OrpaHH'ieHaJIeMHHCKaTOii (x2 + y2)2 = a2(x2 - y2), (x ~ 0);

p(x,y) = XVX2 + y2.

163

3.3.49.D 3aAaHa HepaBeHcTBaMH x ~ 0, y ~ 0, x+y ~ 1, p(x, y) = e Cx+y )2.


3.3.50.	D OrpaHHqeHa KpHBbIMH x 2 = ay, x 2 + y2 = 2a2, y = 0 (x > 0,
	a> 0), p(x,y) = k.
3.3.51.	HafiTH KooPAHHaTbI u;eHTpa TIDKeCTH IIJIaCTHHbI, OrpaHHqeHHOfi na,..
	pa6oJIofi ay = x 2 H npHMofi x + y = 2a, eCJIH nJIOTHOCTb nJIaCTHHhI
	nOCTOHHHa H paBHa Po.
Q CAeJIaeM qepTe:lK		(pHC. 31). HaxoAHM a6cU;HccbI TOqeK A H B nepece--
qeHHH npHMofi x + y		= 2a H napa60JIbI y = ~2 . 11:3 CHCTeMbI YPaBHeHHA

X+ y = 2a

{x2 HaXOAHM Xl = - 2a H X2 = a.

y=a:

Puc. 31

1) Macca nJIaCTHHbI D paBHa

	a	2a-x
m = m(D) = IIPo dxdy = Po	I dx	I dy =
D	-2a	x 2
		a	a	(	x 2 )	9 2	PO·
		= Po	I	2a - X -	a:	dx = 2"a	PO·
			-2a

2) BblqHCJIHM CTaTHqeCKHe MOMeHTbI nJIaCTHHbI OTHOCHTeJIbHO KOOPAH-

HaTHblX ocefi

	a	2a-x
Mx = Po Ilydxdy = Po	I dx	I ydy =
D	-2a	x 2
		a
		=~po J[(2a-x)2-::] dx=356 poa3.
		-2a

164

My = Po IIxdxdy = Po	a	2a-x
My = Po IIxdxdy = Po	I xdx	I dy=
D	-2a	x 2
		a
		= Po Ia ( 2a -	x 2	9 3	Po·
		= Po Ia ( 2a -	X - a )	xdx = -4a	Po·
		-2a
	My	a
Xc = m = -2'					•
OTBeT: Mo (-~, ia).					•

Haii,mu 'Il:oopiJu'ltam'bt qe'ltmpa 1TIJI:HCecmu gJU2YP'bt, 02pa'ltU"te'lt'ltoii, JlU'ltUSlMU:

3.3.52.y = x 2 , Y = 0, X = 4.

3.3.54.x 2 + y2 = R2, Y = 0.

AononHIllTenbHble 3aAaHillSi

3.3.53.	y2 = ax, y = X.
3.3.55.	2	2	2
	x3	+ y3 = a3.

Bbl"tUCJlUmb nJlow,aiJu fjJU2YP, 02pa'ltU"te'lt'lt'btX JlU'ltUSlMU:

3.3.56.

3.3.58.

3.3.60.

3.3.61.

3.3.62.

3.3.63.

Bbl"tUCJlUmb o6t'JeMbl meJl, 02pa'ltU"te'lt'lt'btX n06epX'ltOC1TIJI.MU:

3.3.64. z = 4x2 + 2y2 + 1, Z = 1, x + y = 3, x = 0, y = 0. x 2 y2

3.3.65.z= -+-, z=c.

a2 b2

3.3.66. 3x - 2y = 0, 8x - y = 0, 2x + 3y - 13 = 0, 2x + 3y - 26 = 0,

17x + 16y - 13z = 0, Z = 0.

3.3.67.6x - 9y + 5z = 0, 3x - 2y = 0, 4x - y = 0, x + y - 5 = 0, Z = 0.

3.3.68.Z = 4 - x2 , Y = 5, y = 0, Z = 0.

3.3.69.	Z = a2 -		x 2, X + y = a, y = 2x, y = 0, Z = 0.
3.3.70.	x2	y2	Z2
	2"+2'+2'=1.
3.3.71.	abc
	nJIOCKaJI ImaCTHHa D rrpeACTaBJIlIeT TpeyrOJIbHHK ABC C Bepum-
	HaMH A(I, 1), B(2,2), C(3, 1). nJIOTHOCTb pacrrpeAeJIeHHlI MacC B
	K8.X<AOfi TOQKe pasHa OPAHHaTe 3Tofi TOQKH. OrrpeAeJIHTb
	a) MacCY rrJIaCTHHblj

165

6) CTaUf'leCKUeMOMeHThI IIJIacTUHhI OTHOCUTeJIhHO KOOPAUHaTHhIX

	ocetl:;
	B) KOOPAUHaThI u;eHTpa TIDKeCTU IIJIaCTUHhI.
3.3.72.	Hatl:Tu MacCY IIJIacTUHhI, OrpaHU'leHHotl:KPUBhIMU y = x 2 , Y = v'xl
	eCJIU ee IIJIOTHOCTh paBHa p(x, y) = x + 2y.
3.3.73.	Hatl:Tu MOMeHThI UHepu;uu TpeyrOJIhHUKa ABC C BepWUHaMH
	A(0,1), B(1,2), C(2,1) OTHOCUTeJIhHO KoopAUHaTHhIX ocetl: U Ha-
	'laJIaKOOPAUHaT, eCJIU IIJIOTHOCTh TpeyrOJIhHUKa IIOCTOHHHa H paB-
	HaC.
3.3.74.	Hatl:Tu u;eHTp TIDKeCTU KBaApaTa 0 ~ x ~ 2, 0 ~ y ~ 2 C IIJIOTHo-
	CThIO p(x, y) = x + y.
3.3.75.	Haihu IIJIOW;Mh 'laCTUIIOBepXHOCTU U;UJIUHApa x 2 + y2 = R2, 3a-
	KJIIO'leHHotl:Me:>KAY IIJIOCKOCTHMU Z = mx U Z = nx (m > n > 0).
3.3.76.	Bhl'lUCJIUThIIJIOw;aAh 'lacTUIIOBepXHOCTU U;HJIUHApa x 2 + y2 = ax,
	Bhlpe3aHHotl: U3 Hero cq,epoii x 2 + y2 + Z2 = a2.
3.3.77.	Bhl'lUCJIUThIIJIOW;Mh 'laCTUIIOBepXHOCTU wapa x 2 + y2 + Z2 = a2,
	2	2
	Bhlpe3aHHoii IIOBepXHOCThIO x 2	+ Y2 = l.
3.3.78.	a	b
	Bhl'lUCJIUThKOOPAUHaThI u;eHTpa TIDKeCTU q,urYPhI, OrpaHU'leHHoit
	KapAuouAOii r = a(1 + cos<p).
3.3.79.	HaiiTu Maccy KpYrJIoii IIJIaCTUHhI PMuyca R, eCJIU IIJIOTHOCTh ee
	IIPOIIOPU;UOHaJIhHa KBMpaTY pacCTOHHUH TO'lKUOT u;eHTpa U paBHa
	a Ha KpaIO IIJIaCTUHhI.
3.3.80.	HaiiTu CTaTU'leCKUe MOMeHThI OTHOCUTeJIhHO ocetl: Ox U Oy OA-
	HOPOAHOii q,urYPhI, OrpaHU'leHHOiiKapAuouAOii r = a( 1 + cos <p)
	o~ <p ~ 7r U IIOJIHPHOii OChIO.
3.3.81.	HaiiTu KOOPAUHaThI u;eHTpa TIDKeCTU OAHOPOAHOii q,urYPhI, orpa-
	HU'leHHOiiKPUBhIMU y = X U y2 = ax.
3.3.82.	Hatl:Tu Maccy IIpHMoyrOJIhHOrO TpeyrOJIhHUKa C KaTeTaMU a U b,
	eCJIU ero IIJIOTHOCTh paBHa pacCTOHHUIO TO'lKUOT KaTeTa b.
3.3.83.	Hatl:Tu CTaTU'leCKUeMOMeHThI OTHOCUTeJIhHO oceii Ox U Oy OAHO":
	POAHOtl: q,urYPhI, orpaHU'leHHotl:cUHYCOUAOii y = sin x U IIPHMOii
	oA, IIpOXOAHw;etl: 'lepe3Ha'laJIOKOOPAUHaT U BepWUHY A (i, 1)
	CUHYCOUAhI (x ~ 0).
3.3.84.	Bhl'lUCJIUThMOMeHThI UHepu;uu OTHOCUTeJIhHO KOOPAUHaTHhIX oceii
	TpeyrOJIhHUKa C BepWUHaMU B TO'lKaXA(2, 2), B(0,2), C(2,0).

KOHTponbHble Bonpocbl III 60nee CnO>KHble 3aAaHIIIR

Hai1mu n.l!ow,aau gJUzyP, ozpaHu"teHH:btX '/CpU6'btMU:

3.3.85.x 2 = ay, x 2 = by, y2 = ax, y2 = (3x, a < b, a < (3.

3.3.86.y2 = ax, y2 = bx, xy = a, xy = (3 (0 < a < b, 0 < a < (3).

166

3.3.87.	HaiiTH IIJIOrn;a,Il;b <PHrYPbI, OrpaHHqeHHOii IIPlIMOii	r cos cp = 1 H
	OKppKHOCTblO r = 2 (<pHrypa He CO,LI.ep)KHT IIOJIlOca).
3.3.88.	HaiiTH IIJIOrn;a,Il;b <PHrYPbI, OrpaHHqeHHOii KpHBbIMH r	= a(1-cos cp)

H r = a (BHe Kap,LI.HOH,LI.bI).

3.3.89.BblqHCJIHTb IIJIOrn;a,Il;b IIapa6oJIHqeCKOrO cerMeHTa, OrpaHHqeHHOrO

	X	y)2		X	Y	0	x
	IIapa6OJIoii ( -	+ -	=	- -	- H OCblO		x	.
	a	b		a	b			.
3.3.90.	2az = x 2 + y2, x 2 + y2 + Z2 = 3a2 (BHYTPH IIapa60JIOH,LI.a).
3.3.91.	x 2 + y2 = 2ax, x 2 + y2 = Z2, Z = o.
3.3.92.	B KaKOM OTHomeHHH rHIIep6oJIOH,LI. x 2 + y2 - Z2 = a2 ,LI.eJIHT o6beM
	mapa x 2 + y2 + Z2 ~ 3a2?
3.3.93.	2az = x 2 + y2, x 2 + y2 -			Z2 = a2, z = o.
3.3.94.	HaiiTH 06'beMTeJIa, 3aKJIlOqeHHOrO Me)K,LI.y KOHYCOM

2(x2 + y2) _ Z2 = 0

HrHIIep6oJIOH,LI.OM x 2 + y2 - Z2 = -a2.

3.3.95.BblqHCJIHTb IIJIOrn;a,LI.b qaCTH IIoBepxHocTH U;HJIHH,LI.pa X 2+y2 = 2ax, cO,LI.ep)Karn;eiiclI Me)K,LI.y IIJIOCKOCTblO Oxy H KOHYCOM

x2 + y2 _ Z2 = o.

3.3.96.	HaiiTH IIJIOrn;a,LI.b qaCTH KOHyca z = Jx 2 + y2, «Bblpe3aHHoii»						U;H-
	JIHH,LI.POM (x 2 + y2)2 = a(x2 _ y2).
3.3.97.	BblqHCJIHTb IIJIOrn;a,Il;b qaCTH IIOBepXHOCTH IIapa6oJIOH,LI.a x 2 + Z2 =
	= 2ax, cO,LI.ep)Karn;eiiclI Me)K,LI.y U;HJIHH,LI.POM y = ax H IIJIOCKOCTblO
	x=a.
3.3.98. HaiiTH MOMeHTbI		HHepU;HH		O,LI.HOpO,LI.HOrO	TpeyrOJIbHHKa, orpaHH-
	qeHHOrO IIpllMbIMH x + y =			1, x + 2y ==	2, y = 0,	OTHOCHTeJIbHO
	KOOp,LI.HHaTHblx oceii.
3.3.99. HaiiTH MOMeHTbI		HHepU;HH		O,LI.HOPO,LI.HOii	<PHrYPbI,	OrpaHHqeHHOii
	Kap,LI.HOH,LI.oii r = a(1 + coscp), OTHOCHTeJIbHO oceii Ox, Oy H						OT-
	HOCHTeJIbHO IIOJIlOca.
3.3.100. HaiiTH MOMeHTbI HHepU;HH O,LI.HOPO,LI.HOii <PHrYPbI, OrpaHHqeHHOii 3JI-
	JIHIICOM	x 2	y2
		x 2	y2
		a2	+ b2 = 1,

OTHOCHTeJIbHO oceii Ox, Oy H OTHOCHTeJIbHO Haqana KOOp,LI.HHaT.

3.3.101. HaiiTH MOMeHT HHepu;HH 06JIaCTH, OrpaHHqeHHOii JIeMHHCKaToii

OTHOCHTeJIbHO IIOJIlOca.

3.3.102. HaiiTH CTaTHqeCKHii MOMeHT O,LI.HOpO,LI.HOrO IIOJIYKpyra pa,LI.Hyca R,

JIe)Karn;ero B IIJIOCKOCTH Oxy, OTHOCHTeJIbHO ,LI.HaMeTpa.

167

!(x, y, z)

... ,

§4. TPoViHOVi VlHTErPAl1. CBoVicTBA. BbILfVlCl1EHVlE. npVlMEHEHVlE

OnpE!AeneH ....e TpoMHoro ....HTerpana

Onpe,Il;eJIeHHe TpoitHoro HHTerpaJIa aHaJIOrH'IHOonpe,Il;eJIeHHIO ,Il;BoitHoro HHTerpaJIa. TIYCTb B npOCTpaHCTBeHHoit 06JIaCTH V E 1R3 onpe,Il;eJIeHa H HenpepbIBHa

<PYHKD;HX Tpex nepeMeHHbIX U = !(x, y, z). Pa36HeHHe 06JIaCTH V Ha n npOH3BOJIbHbIX 06JIacTeit ~V1, ~V2, ••• , ~Vn C 06'beMaMH ~V1, ~V2, ~Vn H BbI60p B

KaJK,Il;Oit 06JIacTH ~v i npOH3BOJIbHoit TO'lKHMi n03BOJIXIOT CTPOHTb HHTerpaJIbHYIO

CYMMY BH,Il;a

mn = L !(M;)~Vi.

i=l

Tor,Il;a cym;ecTByeT npe,Il;eJI HHTerpaJIbHbIX CyMM mn npH YCJIOBHH CTpeMJIeHHX K HyJIIO HaH6oJIblIlerO H3 ,Il;HaMeTpOB 06JIaCTeit ~Vi, 9TOT npe,Il;eJI, He 3aBHCXID;Hit OT cnoco6a pa36HeHHX 06JIaCTH V Ha 06JIaCTH ~Vi H BbI60pa TO'leKMi, Ha3bIBaeTCX mpOilH'bI,,M, UHmeepa.llO,M, H 0603Ha'iaeTCXCHMBOJIOM

111!(M}dV H	I I I !(x, y, z) dxdydz.
v	v

TPOitHOit HHTerpaJI 06JIa,Il;aeT CBoitcTBaMH, aHaJIOrH'IHbIMHCBoitcTBaM ,Il;BOitHoro HHTerpaJIa (JIHHeitHocTb, aMHTHBHOCTb, OD;eHKa HHTerpaJIa, CBOitCTBO Cpe,Il;Hero).

Bbl .......cneH....e TpOMHOrO ....HTerpana

TIpe,Il;nOJIO)KHM, 'ITO<PYHKD;HX Tpex nepeMeHHbIX onpe,Il;eJIeHa H He-

npepbIBHa B npOCTpaHCTBeHHoit 06JIaCTH V, KOTopax OrpaHH'IeHacBepxy nOBepx-

HOCTblO z = Z2 (x, y), a	CHH3Y - nOBepXHOCTblO z = Zl (x, y), r,Il;e <PyHKD;HH Zl (x, y)
H Z2(X,y} onpe,Il;eJIeHbI	H HenpepbIBHbI B 06JIaCTH D E Oxy (pHC. 32). Tor,Il;a BbI-

'1HCJIeHHeTpoitHoro HHTerpaJIa CBO,Il;HTCX K nOCJIe,Il;OBaTeJIbHOMY (cnpaBa HaJIeBO) BbI'IHCJIeHHIOonpe,Il;eJIeHHOrO HHTerpaJIa no nepeMeHHoit z (nepeMeHHbIe x H y C'IHTalOTCX npH 3TOM KOHCTaHTaMH) H ,Il;BoitHoro HHTerpaJIa OT Toro, 'ITOnOJIy'lHTCX, no 06JIaCTH D.

Z2(X,y)

111!(x,y,z}dxdydz= IldXdy (			I	!(x,y,Z}dZ).
v	D		Zl (x,y)
B '1aCTHOCTH,eCJIH 06JIaCTb V npe,Il;CTaBJIXeT co6oit npxMoyrOJIbHblit napaJIJIe-
JIenHne,Il;, onpe,Il;eJIXeMblit HepaBeHCTBaMH a ~ x ~ b,				c ~ y ~ d, m ~ Z ~ n, TO
TPOitHOit HHTerpaJI CBO,Il;HTCX K TpeM onpe,Il;eJIeHHbIM HHTerpaJIaM:
	b	d	n
III!(x, y, z) dxdydz = I		dx I	dy I	!(x, y, z) dz.
v	a	c	m

ECTeCTBeHHO, MO)KHO BbI6HpaTb ,Il;pyroit nOpX,Il;OK HHTerpHpOBaHHX.

168

Puc. 32

Puc. 33

3aMeHa nepeMeHHblX B TPOMHOM IIIHTerpane

D;HJIHH,n;pH'ieCKHeKoop,n;HHaTbI r, cp, z (pHC. 33) npe,n;CT8.BJUlIOT co6oil: o606m;e- Hue nOJIHpHbIX Koop,n;HHaT Ha nJIOCKOCTH H CBH3aHbI C npHMoyrOJIbHbIMH KOOp,n;H- HaTaMH x, y, z q,opMYJIaMH

x=rcoscp, y=rsincp, z=z.

IIepexo,n; K TPOil:HOMY HHTerpany B IIHJIHH,n;pHqeCKHX Koop,n;HHaTax ocym;eCTBJIHeTCH lIO q,opMYJIe

!!!f(x,y,z}dxdydz = I!!f(r cos cp, r sin cp, z}rdrdcpdz.

B qaCTHOCTH, eCJIH nOJIO)KHTb B aTOM paBeHCTBe f(x, y, z} == 1, TO nOJIyqHM q,op-

MYJIY ,ll;JIH 06'beMaTeJIa B IIHJIHH,D;pHqeCKHX Koop,n;HHaTax:

v = !!!rdrdcpdz.

C«IleplII'"IeCKllle KOOPAIilHaTbl

Cq,epHqeCKHe Koop,n;HHaTbI r, 8, cp CBH3aHbI C npHMoyrOJIbHbIMH Koop,n;HHaTaMH

x, y, z npH nOMOm;H q,OpMyJI (pHC. 34)

X = rsincpcos8,

{y = rsincpsin8, z=rcoscp.

169

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