Сборник задач по высшей математике 2 том

BU"I,uc./Ium'b UHme2pa./lU, nepeXOOR 71: nO.II.RpHUM 'lCOOpOUHamaM:

3.2.17.	II Ja2 -		X2 -	y2 dxdy, r,ll;e D - nOJIyKpyr X2 + y2 ~ a2, y ~ 0.
	D
	a	~
3.2.18.	I dx	I	Jx2 + y2 dy.
	o	0
3.2.19.	II Ja2 -		x 2 -	y2 dxdy, r,ll;e D OrpaHHqeHa JIeMHHcKaToit

D

3.2.20. II Jl- :: - ~:dxdy, r,ll;e D- BHYTpeHHOCTb 9JIJIHnCa

D

B c./Ieoytow,ux oao'iJ.Hux UHme2pa./laX paccmaaum'b npeoe.llu UHme2pupoaaHUSJ, npUMe'H..R.R np.R.MoY20./l'bHUe U nO.II.RpHUe cucmeMU 7I:OOpOUHam:

3.2.21.	II f(x, y) dxdy, r,ll;e D OrpaHHqeHa OKPY:>KHOCT1.IMH x 2 + y2 = 4x,
	D
	x 2 + y2 = 8x H np1.lMbIMH y = X H Y = 2x.
3.2.22.	II f(x, y) dxdy, r,ll;e D OrpaHHqeHa np1.lMbIMH y = 0, X = 1, y = x.

D

BU"I,uc./lum'b oao'iJ.Hue UHme2pa./lU:

3.2.23.	II(x2 + y2) dxdy, r,ll;e D OrpaHHqeHa KpHBbIMH y = x, X + y = 2a,
	D
	X = 0.
3.2.24.	II ';'x-y-_-y"""2 dxdy, r,ll;e D - TpaneU;H1.I C BepIIIHHaMH A(I,I),
	D
	B(5, 1), C(10, 2), K(2,2).
3.2.25.	II xy dxdy, r,ll;e D OrpaHHQeHa KpHBblMH X + y = 2, x 2 + y2 = 2y
	D
	(x> 0).
3.2.26.	II (X + 2y) dxdy, r,ll;e D OrpaHHQeHa KpHBbIMH y = x 2, Y = ..jX.

D

150

E G)?

j(a'H'H)bte U'Hmeepa.l!'bt 6'bt"tUC.I!Um'b, nepeXOa.H. 11: nO.I!.H.p'H'bt.M. 1I:OOpaU'Hama.M.:

3.2.27.	II J9 - x 2 - y2 dxdy, r,lJ;e D OrpaHH'IeHaKpHBbIMH		y = x, y =
	D
	= V3x, x 2 + y2 = 9.
3.2.28.	II (x2+y2) dxdy, r,lJ;e D OrpaHH'IeHaOKPy:>KHOCTbIO x 2+y2 = 2Rx.
	D
3.2.29.	II dxdy, r,lJ;e D OrpaHH'IeHaJIHHHeii(x 2 +y2)2 = 2ax3 .
	D
3.2.30.	II JR2 - x 2 - y2 dxdy, me D - Kpyr x 2 + y2 ~ Rx.
	D
3.2.31.	II y dxdy, me D - nOJIyKpyr (x - a)2 + y2 ~ a2, y ~ O.
	D
3.2.32.	II (x 2 + y2) dxdy, r,lJ;e D -	Kpyr x 2 + (y + 2)2 ~ 4.
	D
3.2.33.	II arctg ~ dxdy, r,lJ;e D -	'IeTBepTbKpyra x 2 + y2	~ 1, x ~ 0,
	D
	Y ~ O.
3.2.34.	II dxdy, me D OrpaHH'IeHaJIeMHHCKaTOii (x2 + y2)2 = 2a2xy.

D

KOHTponbHble Bonpocbl III 60nee CnO)l(Hbie saACIHIIIR'

3.2.35. qTO Bblprur<aeT 3HaK 51Ko6HaHa npeo6pa30BaHH5I KOOp,lJ;HHaT?

3.2.36. IIo'IeMYnpH npeo6pa30BaHHH KOOp,lJ;HHaT B ,lJ;BoiiHOM HHTerpaJIe Heo6xo,lJ;HMa B3aHMHM O,lJ;H03Ha'IHOCTb9Toro npeo6pa30BaHH5I?

3.2.37. IIo'IeMY<PyHKIJ;HH x = x(u,v), y = y(u,v) HCnOJIb3yeMble npH 3aMeHe nepeMeHHbIX B ,lJ;BoiiHOM HHTerpaJIe ,lJ;OJDKHbI 6bITb ,lJ;H<p<pepeHIJ;HpyeMbIMH (npH (u,v)

3.2.38. MO)l<HO JIH BbmOJIHHTb TaKoe npeo6pa30BaHHe, 'ITo6bI COOTBeTCTByIOIu;HM HHTerpaJIOM BbI'IHCJIHTb,lJ;JIHHy KpHBOii?

3.2.39. IIpH COCTaBJIeHHH nOBTopHoro HHTerpaJIa nOJIY'IHJIaCb3anHCb

X 3x 2

Idx I f(x,y)dy.

a 2x-y

KaKoii 06JIaCTH D MO)l{eT COOTBeTCTBOBaTb 9TOT HHTerpaJI?

B aa'H'H'btx a60il.'H'btX U'Hmeepa.l!ax nepeil.mu 11: no.I!.H.p'H'bt.M. 1I:OOpaU'Hama.M. U pacCma6Um'b npeae.l!'bt U'HmeepUp06a'HU.H.:

3.2.40. II f(x, y) dxdy, D - Kpyr x 2 + y2 ~ ax.

D

151

3.2.41.	JJI(x, y) dxdy, 06JIaCTb D - o6rn;ruI 'IaCTbKpyroB x 2 + y2 ::;; ax,
	D
	x 2 + y2 ::;; bx.
3.2.42.	JJI(x, y) dxdy, 06JIaCTb D OrpaHH'IeHaIIpHMbIMH y = -x, y = x,

D

y=l.

B aa'lt'H!btX UHme2pa.ll.aX npOU36ecmu Y'ICa3aHHY'lO 3aMeHY nepeMeHH'btX U pacCma6Um'b npeae.ll.'bt UHme2pUp06aH'U.R.:

	b	/3x
3.2.43.	Jdx	JI(x, y) dy (a> 0, a > 0), eCJIH U = x, v = ~.
	a	ax
	1	1
3.2.44.	Jdx	JI(x, y) dy, eCJIH U = x + y, V = X - y.

o 0

3.2.45.JJI(x, y) dxdy, me D - 06JIaCTb, OrpaHH'IeHHaHKPHBOii

D

(X2 + ly2)2 = x 2y,

		eCJIH x = r cos <p, y = V3r sin <po
	3.2.46.	JJI(x, y) dxdy, r,!l;e D OrpaHH'IeHaIIapa60JIaMH y = ax2, y = by2
		D				eCJIH y = ux2, xy = v (0 < a < b,
		H rHIIep60JIaMH xy = p, xy = q,
		0< p < q).
	3.2.47.	TIpeo6pa30BaTb C IIOMOlII,bIO IIO,!l;CTaHOBOK x = ar cos <p, y = br sin <p
		HHTerpaJI
				JJ1(R2 - :~ -		~:) dxdy,
				D
		r,!l;e D - JIe)Karn;ruI B IIepBoii 'IeTBepTH 'IaCTb 9JIJIHIITH'IeCKOrO
		KOJIbIJ;a
	JIepe11mu	'IC nOJlJlpH'bt.M		'lCOOpaUHamaM (r, <p) U pacCma6Um'b npeae.n.'bt UHme-
	2pUp06aH'U.R. 6 mOM U 6			apY20M nopSla'ICe 6		aaHH'btX UHme2pa.ll.ax:
		1	1				1	~
	3.2.48.	Jdx	JI(x, y) dy.		3.2.49.		J	J I(x,y) dy.
							dx
		0	0				0	i-x
		2	xv'3
	3.2.50.	J	JI( Jx 2 + y2) dy.
		dx
		0	x

152

3.2.51.B ,!l;BOtiHOM HHTerpaJIe

IIf(x,y)dxdy,

D

	r,!l;e 06JIaCTb D OrpaHH'IeHaKpHBbIMH ..;x+vY = va, x = 0, y = 0,
	C,!l;eJIaTb 3aMeHY nepeMeHHblx x = U cos4 V, Y = U sin4 v. HHTerpaJI
	npHBeCTH K nOBTopHOMY.
3.2.52.	BbI'IHCJIHTb	II dxdy,
		D
		x2	y2	X	Y
	r,!l;e D OrpaHH'IeHaKPHBOti a2		+ b2 = Ii		+ k·
3.2.53.	BbI'IHCJIHTb	II dxdy,
				Vf = 1, x = 0, y = o.
		D	VI +
	r,!l;e D OrpaHH'IeHaKpHBbIMH
3.2.54.	BbI'IHCJIHTb	II r21sin (<p + ~f) - rl
				drd<p,
		D
	r,!l;e D - npHMoyrOJIbHHK 0 ::;; r ::;; 1,			0::;;	<p ::;; 21f.

§ 3. np~MEHEH~SI ABOIf1HOrO ~HTErpAJlA

.II:Boil:Hble HHTerpaJIbI HCIIOJIb3YIOTCH IIpH peIIIeHHH MHorHX reOMeTpH'IeCKHXH <pH3H'IeCKHX3a,II;a'l:BbI'IHCJIeHHHllJIOIrrap;eil: llJIOCKHX <pHryp H 1I0BepxHocTeil:, 06'b- eMOB TeJI, Koopp;HHaT rreHTpa TH:lKeCTH, MOMeHTa HHeprrHH H T. p;.

1. ECJIH D - OrpaHH'IeHHaH06JIacTb llJIOCKOCTH Oxy, TO ee llJIOIIIa,II;b S BbI-

QHCJIHeTCH 110 <p0pMYJIe

	S = S(D) = Ildxdy.
	D
2. TIYCTb Z = I(x, y) -	HeoTPHrraTeJIbHaH, HellpepbIBHaH <pYHKrrHH B 3aMKHY-
Toti 06JIacTH D. ECJIH V -	TeJIO, OrpaHH'IeHHOecBepxy 1I0BepXHOCTbIO Z = I(x,y),
CHH3Y - OMaCTbIO D, a	C60KY - COOTBeTCTBYIOIIIeil: rrHJIHHp;pH'IeCKoil: 1I0Bepx-

IiOCTbIO C 06pa3yIOIIIeil: lIapaJIJIeJIbHoil: OCH Oz H HallpaBJIHIOIIIeil:, COBlla,II;aIOIIIeil: C rpaliHrreil: 06JIacTH D, TO 06'beM3TOro TeJIa paseH

V= 111(x, y) dxdy.

D

3. TIYCTb V - TeJIO, OrpaHH'IeHHOecBepxy 1I0BepXHOCTbIO Z = I(x, y), CHH3YnoaepxHocTbIO Z = g(x, y), IIpH'IeMIIpoeKIIHeil: 06eHx 1I0BepxHocTeil: Ha llJIOCKOCTb

153

Oxy CJIY:lKHT 06JIacTb D, B KOTOPOit <PYHKD;HH f(x,y) H g(X,y) HenpephIBHbI (H f(x,y) ~ g(X,y)), TO 06'beM9TOm TeJIa paBeH

v = /JU(X,y) -g(x,y))dxdy.

D

4. IIyCTb nOBepXHOCTb 3ap;aHa YPaBHeHHeM Z = f (X, y), (X, y) ED, rp;e <PYHK- D;H:!I f (x, y), a TaK:lKe ee '1aCTHblenpoH3Bop;HbIe nepBoro nOp:!lp;Ka, HenpepbIBHbI B 06JIacTH D. Torp;a ee nJIOIII~ S BbI'IHCJI:!IeTC:!Ino <p0pMYJIe

S= //Jl + f~2(X,y) + f~2(X,y)dxdy.

D

IIpHH:!ITbI TaK:lKe 0603Ha'leHH:!I:f~(x, y) = p, f;(x, y) = q. B TaKOM CJIY'Iae,

S = //..)1 + p2 +q2 dxdy.

D

Bbl,·..tCneHllle cl>1II3I11'"1eCKIIIX III MeXaHIII'"IeCKIIIX Benlll'"llllH

IIpep;noJIO:lKHM, 'ITOIIJIOCKaH nJIacTHHa D HMeeT nOBepXHOCTHYIO IIJIOTHOCTb

pacnpep;eJIeHH:!I Macc p(x,y) HenpepbIBHYIO B D. Torp;a Macca m = m(D) 9TOit nJIacTHHbI BbI'IHCJI:!IeTC:!Ino <popMYJIe

m= //p(x,y)dxdy

D

(<pH3H'IeCKHitCMbICJI ,lI;BoitHoro HHTerpaJIa).

MOMeHTbI HHepD;HH Jx , Jy H Jo nJIOCKoit MaTepHaJIbHoit IIJIacTHHbI D C nOBepxHOCTHOi! IIJIOTHOCTbIO p(x,y) OTHOCHTeJIbHO Koopp;HHaTHbIX oceit Ox, Oy H Ha'laJIa Koopp;HHaT 0(0,0) COOTBeTCTBeHHO BbI'IHCJI:!IIOTC:!Ino <popMYJIaM:

Jx = //y2 p(x,y)dxdy ; J y = //x 2p(x,y)dxdy;

D		D
J o = Jx +Jy	= //(x 2 +y2)p(x,y)dxdy.
	D
B CJIY'IaeOP;HOPOP;HOit IIJIacTHHbI (p = 1) 9TH <P0PMYJIbI npHHHMaIOT 60JIee
npoCToit BHP;:
Jx = //y2 dXdY , J y =	//x 2 dxdy,	Jo = / /(x 2 + y2) dxdy.
D	D	D

KoopP;HHaTbI D;eHTpa T:!I:lKeCTH MaTepHaJIbHoit nJIaCTHHbI D C nJIOTHOCTbIO p(x,y) BbI'IHCJI:!IIOTC:!Ino <popMYJIaM

Mx

YC=m'

154

me

My = !!xp(x,y)dxdy,	M", = !!yp(X,y)dXdy-
D	D

CTaTH'IeCKHeMOMeHTbI nJIaCTHHbI D OTHOCHTeJIbHO oceit Ox H Oy COOTBeTCTBeHHO, a m - ee Macca.

B CJIy'laeO)J;HOPO)J;HOit nJIacTHHbI COOTBeTCTBeHHO HMeeM:

!!XdXdy	!!ydXdy
D	D
Xc = ----:,-:---	Yc = ----:,-:---
!!dXdy'	!!dXdy
D	D

3.3.1.BbPUlCJIHTh IIJIOrn;a,rr.h <PHrYPhI, OrpaHH'IeHHotiKpHBhIMH y2 = 2x H

y=x.

o MMeeM S =!!dxdy. HanpaBJIeHHe, HJIH nOpH,IIOK, HHTerpHpOBaHH5I BhI-

D

6epeM TaK, KaK YKa3aHo Ha 'IepTe)Ke(pHC. 24).

			y	x=y
			2
			o	2
				Pu.c. 24
CHa'IaJIaonpe,IJ;eJIHM KOOp,IJ;HHaThI TO'IKHA:
{	y2 = 2x	2	= 2x	=> Xl = 0, YI = °H X2 = 2, Y2 = 2.
	=> X
	y =x,

npOeKlIH5I 06JIacTH D Ha OCh Oy eCTh oTpe30K [0,2]. TaKHM 06Pa30M,

3.3.2. BhI'IHCJIHThnJIOrn;a,rr.h napa60JIH'IeCKOrOcerMeHTa AOB, orpaHH-

'IeHHOrO,IJ;yroti BOA napa60JIhI y = ax2 H OTpe3KOM BA, COe,IJ;H- H5IlOrn;HM TO'IKHB(-1,2) H A(1,2).

155

a £CHO, 'ITOypaBHeHHe IIapa60JIbI HMeeT BH)]; y = 2X2 (y(-l) = y(l) = 2).

<l>Hrypa D, IIJIOm;a)l;b KOTOpoii Ha)l;O BbIqHCJIHTb, OrpaHHqeHa CHH3Y IIapa6oJIoii y = 2x2 , a cBepxy - IIPHMOii Y = 2. CJIe)];OBaTeJIbHO,

3.3.3.	BbIqHCJIHTb IIJIOm;a)l;b IIeTJIH KPHBOii
	( X2 + y2)2 = 2xy .
	a2	b2	e2

a nO)]; IIeTJIeii 6y)];eM IIO)];pa3YMeBaTb 06JIacTb, OrpaHHqeHHYIO )];aHHOii KpHBoii H paCIIOJIO)l{eHHYIO B IIepBoii qeTBepTH (x ~ 0, y ~ 0). BOCIIOJIb3yeMcH o606m;eHHbIMH IIOJIHPHbIMH KOOp)];HHaTaMH: x = a . r cos <p, y = b . r sin <po B

TaKOM cJIyqae, HKo6HaH IIpeo6pa30BaHHH paBeH

		ax	ax
	J=	or	o<p	= lac?s<p		-ar sin <pI = a. b· r.
		oy	oy	bsm<p		brcos<p
		or	o<p
KPHBaH B IIOJIHPHbIX KOOp)];HHaTaX HMeeT BH)];
	2	2		2 . 2)2	=	2abr2 sin <pcos <p	'
	( rcos<p+rsm<p					2
T. e. (r2)2 =	abr2 . 2 sin <p cos <p					...;ab .
		2		' oTKy)];a r		= -e-y'sm2<p. BHYTpeHHocTb IIe-

e

TJIH, T. e. 06JIaCTb HHTerpHpoBaHHH D B IIpHMoyrOJIbHbIX KOOp)];HHaTaX, 3a-

)];aeTCHHepaBeHCTBOM
(	X2		y2)2	2xy
	a	2	2	~ 2·
			+ b

B IIOJIHPHbIX KOOp)l;HHaTaX COOTBeTcTByIOm;aH 06JIacTb HHTerpHpOBaHHH G

OIIpe)];eJIHeTCH HepaBeHCTBOM 0 ~ r					~ ~y'sin2<p, IIpH 9TOM sin 2<p ~ 0, T. e.
o~ <p ~ ~. TaKHM 06Pa30M,
s = II dxdy = II abrdrd<p =
D	G	Vab =
	11"				11"		Vab
	-	-ySlD2cp			-
	2	c	I		2	(21-oVSID2CP)
	= ab I d<p			r dr = ab I d<p		r2 0 c
	o		0		0
			11"				11"
			"2		2	2	"2	2
		= at I		~~sin 2<pd<p = ~;			I sin 2<p d<p = (~~)	. •
			o				0

156

B'bt"tuc.!tum'b n.!tow,aau rjjuzyp, ozpa'Hu"te'H'H'btX ?CPU6'btMU:

3.3.4.x = 0, y = ~X, Y = 4 - (x - 1)2.

3.3.5.xy = 4, x + y - 5 = O.

3.3.6. ..;x + fY.... = Va, x + y = a.

3.3.7.x 2 +y = ax, y2 = 2ax, x = 2a, y ~ O.

3.3.8.y2 = lOx + 25, y2 = -6x + 9.

3.3.9.x 2 + y2 = 2x, x 2 + y2 = 4x, Y = x, Y = O.

3.3.10.(x 2 + y2) = 2ax3 , a > O.

3.3.11.x 2 + y2 + 2y = 0, Y = -1, Y = -x.

3.3.12.BbI'UfCJUlTbIIJIOrn;a,n;b qmrYPbI, OrpaHHqeHHoil: KPHBOil:

(4x - 7y + 8)2 + (3x + 8y - 9)2 = 64.

a BbIQHCJIeHH5I IIO <p0pMYJIe

s = II dxdy

D

HeIIpHeMJIeMbI BBH.n:y CJIO)l{HOCTH IIpe,n;eJIOB HHTerpHpOBaHH5I. llpoH3Be,n;eM 3aMeHY IIepeMeHHblx IIO <p0pMYJIaM

						1
{ 4X -7y + 8 = u					X = 53 (8u + 7v + 1)
				{
3x + 8y -	9 = v,				Y =	l3 (-3u + 4v + 60).
ax 8	ax	7	ay	3	ay	=	4
TIPH 3TOM au = 53'	av	= 53'au		= -53'	av		53' T.e.
		ax	ax	8		7
		au	av	53		53	1
	J =	ay	ay	3		4	53 .
		au	8v	53		53

B IIJIOCKOCTH Koop,n;HHaT (u, v) COOTBeTCTBYIOrn;M JIHHH5I HMeeT BH,n; u 2 +v 2 =

== 64, T. e IIpe,n;CTaBJI5IeT co6oil: OKPY)I{HOCTb, a 06JIacTb G - Kpyr u2 +v2 ~ 64

C IIJIOrn;a,n;bIO S(G) = 641r.			MCIIOJIb3Y5I COOTBeTcTByIOrn;He <P0PMYJIbI,				IIOJIY-
'laeM	S = II dxdy = II J dudv = II 53 dudv = 53 S (G) = 6:;.
							•
					1	1
	D	G			G
3.3.13.	(x + y - 1)2	(x - Y + 3)2
	4	+	9		=1.
3.3.14.	(2x + 3y - 5)2		(3x -	2y + 1)2
	16	+		25	= 1.
3.3.15. BblQHCJIHTb IIJIOrn;a,n;b <PHrYPbI, OrpaHHQeHHoil: KpHBbIMH
	r = a(1 + COSip),			r	= a COS ip,	(a> 0).

157

2a

Puc. 25

a JIHBlBr )];aHbI B nomlpHbIX KOOp)];HHaTax, n09TOMY BOCnOJIb3yeMC5I <pOp-

MYJIoA nJIOIIIaAH B nOJI5IpHbIX KOOp)];HHaTax

8 = IIrdrd<p.

G

llepBM <PYHKIJ;H5I r	= a(1 + cos <p) onpe)];eJIeHa npH <p E [-71", 71"1, a BT0-
pM r = a cos <p -	npH <p E [- ~, ~], TaK KaK npH npOqHX 3HaQeHH5IX <p
nOJIYQaeTC5I r < O.	COOTBeTCTBYIOIIIM 06JIacTb HMeeT BH)];, H306p8JKeHHbIA

Ha pHC. 25. BBH)l;y CHMMeTpHH <PHrYPbI OTHOCHTeJIbHO nOJI5IpHoti OCH MO)l{- HO OrpaHHQHTbC5I BbIQHCJIeHHeM nOJIOBHHbI nJIOIIIaAH, a pe3YJIbTaT Y)];BOHTb. lIMeeM

11"	a{l+cos<p)				a{l+cos <p)
8=2 I"2				11"
	d<p	I	rdr + 2	I d<p	I	rdr =
o	acos <p			11"	o
				"2
		11"
		"2			11"
	= a2	I (1 + 2 cos <p) d<p + a2 1(1 + 2 cos <p + cos2 <p) d<p =
		o			11"
					"2
= a' [1(1 + 2 cos,,) d<p +1(1+ 200',,) d<p +1cos' "d,,]
				2		2
		= a' [i(1 + 2=,,) d<p +1(1+ ,;"2,,) d<p] = ~~a'. •
						2

B'bI."tuc.lI:um'b n.l/,O'l4aaU rjjueyp, oepa'Hu"te'H'H'bI.X "'PU6'b1.MU:

3.3.16.(x42 + y:) 2 = x; _Y:.

3.3.17.(y - X)2 + x 2 = 1.

3.3.18.x 3 + y3 = 2xy, x ~ 0, y ~ O.

158

3.3.19. x 2 + y2 = 2ax, x 2 + y2 = 2bx, y = 0, y = x, 0 < a < b. x 2 y2

3.3.20.a2 + b2 = 1.

3.3.21.xy = a2, xy = b2, Y = m, Y = n (a> b; m > n).

3.3.22.BbI9HCJIHTb 06'beMTeJIa, orpaHHgeHHoro nOBepxHocTflMH y = y'x,

y = 2y'x, x + z = 4, Z = o.

o IIepBble ,n;Ba ypaBHeHHfI H306prur<aroT napa60JIHgeCKHe IJ;HJIHH,n;PbI C BepTHKaJIbHOti o6pa3yrorn;eti, TpeTbe,T. e. x + Z = 4 - ypaBHeHHe HaKJIOHHOti nJIOCKOCTH, a ypaBHeHHe z = 0 - nJIOCKOCTb Oxy. CooTBeTcTByrorn;ee TeJIO H306prur<eHo Ha pHC. 26; cBepxy ero orpaHH9HBaeT nOBepXHOCTb Z = 4 - x.

z

y

				4	x
	Puc. 26	Puc. 27
06'beMTeJIa BbI9HCJIHM no <p0pMYJIe
		V = / / (4 - x) dxdy,
		D
r,n;e 06JIaCTb D H306prur<eHa Ha pHC. 27. IIMeeM
V = /(4	2,rz	2,rz
	-x) dx / dy =	/(4 -x) dXYI,rzx =
o	,rz	0
		4	~x~) 14 = 128			•
	= /(4 - x)v'x dx = (4. ~x~ _
		o	5	0	15·
		3
3.3.23.	BbI9HCJIHTb 06'beMTeJIa, orpaHHgeHHoro nOBepXHOCTflMH Z = 0,

Z = 2 - y, y = x 2 •

Q TeJIo, 06'beMKOToporo Hy:>KHO BbI9HCJIHTb, H306prur<eHo Ha pHC. 28. B CHJIy cHMMeTpHH TeJIa (KJIHHa) OTHOCHTeJIbHO nJIOCKOCTH Oyz, BbI9HCJIHM 06'b-

eM nOJIOBHHbI TeJIa H pe3YJIbTaT y,n;BOHM. Koop,n;HHaTbI T0geK A H B y,n;OBJIeTBOPflroT CHCTeMe ypaBHeHHti y = x 2 H Y = 2, oTKy,n;a A(J2, 2), B(-J2,2).

159