Сборник задач по высшей математике 2 том

5. ,I1;.rrH JIIo6oil: <PYHKU;HH f(x, y), HenpephIBHoil: Ha 06JIaCTH, D HMeeT MeCT() HepaBeHCTBO

If!f(x,Y)dXdyl ~!!If(x,y)ldXdy.

D D

Bbl'"lMCneHMeABOMHOrO MHTerpana

IIpe)l:noJIO)KHM, 'ITO06JIacTb D MO)KHO 3a)l:aTb B BH)l:e CHCTeMhI HepaBeHCTB:

reOMeTpH'IeCKH 9TO 03Ha'iaeT, 'ITO K~)l:aH BepTHKaJIbHaH npHMaH X = Xo

(a < Xo < b) nepeceKaeT rpaHHIJ;y 06JIaCTH D TOJIbKO B )l:BYX TO'lKaxMl H M2

(pHC. 11), KOTopble Ha3hIBaIOTCH COOTBeTCTBeHHO TO'lKOil:BXO)l:a H TO'lKOil:BhIXO)l:a.

Tor)l:a

b Y2(X)

!!f(x,y)dXdy=!dX! f(x,y)dy.

D a yt{x)

y

X

Puc. 11

Puc. 12

ECJIH )Ke 06JIaCTb D (pHC. 12) MO)KHO 3a)l:aTb B BH)l:e CHCTeMhI HepaBeHCTB:

TO

	d	X2(Y)
!!f(x,y)dxdy = ! dy		!	f(x,y)dx.
D	c	xt{y)

HHTerpaJIhI, CTOHIIJ;He B npaBhIX '1acTHXnpHBe)l:eHHhIX paBeHCTB, Ha3hIBaIOTCH nOBmOp'H'MMU (HJIH .D:BYKpaTHhIMH). OHH OTJIH'IaIOTCH)l:pyr OT )l:pyra nOpH)l:KOM HHTerpHpOBaHHH. HHTerpaJI, CO)l:ep)KaIIJ;HiI: <PYHKU;HIO f(x, y), Ha3hIBaeTCH B'ltympe'H- 'ltUM, )l:pyroil: - B'lteW'ltUM. IIpH BhI'IHCJIeHHHnOBTopHhIX HHTerpaJIOB CJIe.n:yeT 6paTb CHa'laJIaBHyTpeHHHiI: HHTerpaJI, npH 9TOM nepeMeHHali, He CTOHIIJ;aH nO)l: 3HaKOM

130

13).

,n1i<P<pepeHIJ;liaJIa, npliHIiMaeTCSI nOCToSlHHOit. 3aTeM BhPllieJISleTCSI BHemHHit HHTerpaJI (TaKIiM 06pa30M, IiHTerpHpOBaHHe B nOBTOpHOM HHTerpaJIe H,D;eT CnpaBa HaJIeBO). K3JK,D;hlit H3 HHX BhI'IHeJISleTCSInpH nOMOmH <P0PMYJIhI HhIoToHa-JIeit6HHIJ;a, KaK Onpe,D;eJIeHHhlit HHTerpaJI.

06JIacTH, He npe,D;CTaBHMhle B onHcaHHOM Bhlme BH,D;e, eJIe,nyeT pa36HTh Ha KoJIe'lHOe'1HeJIOTaKHX 06JIaCTeit npH nOMOmH npSlMhlX, napaJIJIeJIhHhlX KOOp,D;HHaT-

JIbIM OCSIM (CM. pHC. IIpH BhI'IHCJIeHHH,D;BOitHhIX HHTerpaJIOB no TaKHM o6JIa-

CTSIM eJIe,nyeT npHMeHHTh CBOitCTBO a,n,nHTHBHOCTH (CBOitCTBO 4).

y

o

x

Puc. 13

Puc. 14

3.1.1.On;eHHTh HHTerpaJI

	!!(x + y -	5) dxdy,
	D
r,n:e 06JIacTb HHTerpHpOBaHHH D - 9TO Kpyr x 2 + y2 ~ 16.
Q Heo6xo,n:HMO	HaihH HaH60JIbIIIee	H HaHMeHbIIIee 3Ha'leHHH <PYHKn;HH
J(x, y) = x+y -	5 Ha Kpyre x 2+y2 ~ 16 H rrpHMeHHTb on;eHKy H3 cBoiicTBa 2.

<PYHKn;HH Z = x +y rrpHHHMaeT 3Ha'leHHe0 Ha rrpHMoit x +y = O. Ha rrpHMbIX x + y = C, rrapaJIJIeJIbHbIX rrpHMoit x + y = 0, <PYHKn;HH Z rrpHHHMaeT 3Ha'leHHe C. CJIe,n:oBaTeJIbHO, <PYHKn;HH Z = x + y (a 3Ha'lHT,H <PYHKn;HH J(x,y)) rrpHHHMaeT Ha Kpyre MaKCHMaJIbHOe 3Ha'leHHeB TO'lKeM (2V2, 2V2) (CM. pHC. 14) H MHHHMaJIbHOe 3Ha'leHHe- B TO'lKeN( -2V2, -2V2). IIpH 9TOM

HMeeM J(M) = 4V2-5 H f(N) = -4V2-5. IIoCKOJIbKY rrJIOIn;a,D;b Kpyra paB-

Ha 7rR2 = 167r, TO COrJIacHo CBOitCTBY 2 ,n:BoitHoro HHTef'paJIa(m = -4V2 -	5
H M = 4V2 - 5), rrOJIY'laeM
-167r(4V2 + 5) ~ !!(x + y - 5) dxdy ~ 167r(4V2 - 5).	•
D

131

3.1.2.On;eHHTb HHTerp8.JI

	!!(4x2 +y2 -	2)dxdy,
	D
a	r.n.e 06JIaCTb HHTerpHpOBaHHH D -		Kpyr X 2 + y2 ~ 16.
	TaK KaK 4x2 + y2 - 2 ~ 0, TO On;eHKa CHH3Y 4x2 + y2 - 2 ~ -2, V(x, y) E
E ]R2 OqeBH.n.Ha. IIo9ToMY MO:lKHO IIpHHHTb m			= -2 = 1(0,0), r.n.e I(x,y) =
= 4x2 + y2 - 2. l..JTo6bI BblqHCJIHTb M =		max	1(x, y), BOCIIOJIb3yeMcH IIapa-
	(x,y)ED

MeTpHqeCKHMH ypaBHeHHHMH OKPY:lKHOCTH: x = 4cost, Y = 4sint, t E [0,2nj. Tor.n.a IIpH JII060M t

1(4cost,4sint) = 64cos2 t + 16sin2 t -				2 =
	= 16(sin2 t + cos2 t) + 48 cos2 t -					2 = 48 cos2 t + 14 ~ 62,
T. K. cos2 t ~ 1. BMecTe C 9THM I(x, y) IIpHHHMaeT 3HaqeHHe M = 62 IIpH
t = 0,	T.e. M = 1(4,0) = 62. OTclO.n.a,			yqHTbIBaH, qTO IIJIOIIIa,II.b S Kpyra
x 2 + y2 ~ 16 paBHa 16n, IIOJIyqaeM on;eHKy
	-32n ~ !!(4x 2 + y2 -			2) dxdy ~ 992n.					•
			D
O'4eHUm'b UHme2pa.!tu:
3.1.3.	!!(x + y + 1) dxdy, r.n.e D -			Kpyr x 2 + y2 ~ 4.
	D
3.1.4.	x + 2y -		1			x 2		2
	IfCOS x	2 + 3y2	+ 2 dxdy, r.n.e	D -	9JIJIHIIC -3 + Y				~ 1.
	D				IIpHMOyroJIbHHK °~ x ~ 1,
3.1.5.	!!(x + xy - x 2 - y2) dxdy, r.n.e D -
	D
	°~ Y ~ 2.
3.1.6.	!!(x 2 -	y2) dxdy, r.n.e D - Kpyr x 2 + y2 ~ 2x.
	D				= !7 dx !x 2
3.1. 7.	BhlqHCJIHTb IIOBTOPHblit HHTerp8.JI I						dy.
					o	0
a CHaq8.JIa BblqHCJIHM BHYTpeHHHit HHTerp8.JI IIO <popMYJIe HbIOToHa-JIeit6-

HHn;a. Ero pe3YJIbTaT 6y.n.eT IIo.n.bIHTerp8.JIbHoit <pYHKn;Heit ,II.JlH BHeIIIHero HH-
Terp8.JIa.	,
			•
	3	3x	,
3.1.8.	BblqHCJIHTb IIOBTOPHblit HHTerp8.JI I = ! dx	!	~ dy.
	1	x

132

o MH0:lKHTeJIh l (OH He 3aBHCHT OT y, rr09TOMY M0:lKeT C'IHTaThCfIrrOCTOflHflbIM ,n;JIfI BHYTpeHHero HHTerpaJIa) MO:>KHO BhIHeCTH 3a 3HaK HHTerpaJIa, T. e. nepeHeCTH BO BHeIIIHHil: HHTerpaJI:

Bbt"lUC.I/,Umb nOfJmOpH'bte UHme2pa.l/,U:
	!a dx		!Vx dy.				2	Iny
3.1.9.						3.1.10.	! dy	!		eX dx.
	o		0				1	0
3.1.11.	!2 dy !1			(x 2 + 2y) dx.		3.1.12.	!4 dx !2		(x + y)-2dy.
	o	0					3	1
3.1.13.	!3	dy		!5	(x + 2y) dx.
	-3		y2-4
3.1.14.	BhI'IHCJIHTh,n;BOil:HOil: HHTerpaJI 1= !! 1 +X2							2 dxdy, r,n;e D - rrpfl-
							D	y

MoyrOJIhHHK 0 ~ x ~ 2, 0 ~ y ~ 1.

Q IIpe06pa3yeM ,n;BOil:HOil: HHTerpaJI B rrOBTopHhlil:. IIpe,n;eJIhI HHTerpHpOBaHHfI H3BeCTHhI, rr09TOMY

2	2	!1	dy	x312	11 8 'If 2
1= ! x	dx·		1+y2	=3 o·arctgY. o=3"4" = 3'If.
o		0

IIoBTopHhlil: HHTerpaJI CBeJICfI K rrpOH3Be,n;eHHIO ,n;ByX He3aBHCHMhIX ,n;pyr OT ,n;pyra orrpe,n;eJIeHHhIX HHTerpaJIOB, rrOCKOJIhKY pe3YJIhTaT BhI'IHCJIeHHfIBHy-

TpeHHero HHTerpaJIa eCTh 'IHCJIO.		•
3.1.15. BhI'IHCJIHTh,n;BOil:HOil: HHTerpaJI I = If	y dxdy /' r,n;e D -
D	(1 + x 2	+ y2)3 2

KBa.n;paT 0 ~ x ~ 1, 0 ~ y ~ 1.

Q ,I1;aHHhlil: ,n;BOil:HOil: HHTerpaJI MO:>KHO rrpe,n;CTaBHTh B BH,n;e rrOBTopHoro ,n;By-

Mfl crroc06aMH:

!	1 d	!1	ydy	HJIH !	1
o		x 0	(1 + X2 + y2)3/2	o

1
ydy!	dx
0	(1 + X2 + y2)3/2 .

BH3YaJIhHOe Ha6JIIO,n;eHHe rrOKa3hIBaeT, 'ITOrrpOm;e 6paTh rrepBhlil: HHTerpaJI,

TaK KaK ero BHYTpeHHHil: HHTerpaJI JIerKO CBO,n;HTCfI K Ta6JIH'IHOMY.TaKHM

133

06Pa30M, CqHTaeM nepBbltl: HHTerpaJI:

1- j1 dx j1 1 d(1 + x 2 + y2)			_ j1 (_	1	11) _
- 0 0 2 (1 + x 2 + y2)3/2			- 0	Jl + x 2 + y2	0 -
	1	~)dX = [In(x+ Jl +x2) -In(x+ J2+X2)]11=
= j(k -
o	l+x2	2+x2			0
		= In(1 + V2) -In(1 + V3) + In V2 = In 2 + v;. •
					. 1 +v3

B'bt"tuc.I!um'b aeoi1:no1i U'Hme2pa.l! no aa'H'Ho1i o6.1!acmu D:

3.1.16.	jj xydxdy, r,n;e D: 0 ~ x ~ 1, 0 ~ y ~ 2.
	D
3.1.17.	If		dxdy	2'r,n;e D: 0 ~ x ~ 1, 0 ~ y ~ 1.
	D	(x + y + 1)
3.1.18.	If	x dxdy				. J7)
		2	2 'r,n;e D: 0 ~ x ~ 2,		x ~ y ~ Xv 3.
	D	X	+y
3.1.19.	If	dxdy		2'r,n;e D: 1 ~ x ~ 3, 2 ~ y ~ 5.
	D	(x + 2y)
3.1.20.	BblqHCJIHTb HHTerpaJI I = If				xdxdy
					2	2'r,n;e 06JIaCTb D - napa6o-
				D	x	+y
	JIHqeCKHtl: cerMeHT, OrpaHHqeHHbltl: napa60JIotl: y = ~X2 H npHMotl:
	y=x.

Q Iho6pa3HM 06JIacTb HHTerpHpOBaHHH D (pHC. 15). TaK KaK npHMM y = x

H napa60JIa y = ~x2 nepeceKalOTCH B TOqKaX 0(0,0) H A(2, 2), TO 06JIacTb

D onpe,n;eJIHeTCH CHcTeMotl: HepaBeHCTB {ox2~ x ~ 2,

"2 ~ y ~ x.

y

2 ----------- I A

,,' :

o

I

I

I

I

I

I

: x

2

Puc. 15

134

Tenepb BblqHCJIHM HCKOMblii HHTerpaJI I:

2

x

dy

/2

(1

y IX

)

1= /

xdx

/

x 2 + y2

=

o

x dx

x arctg X x2

=

o

x2

2

2"

2

2

2

2

= /

(arctg 1 - arctg ~) dx = ~ / dx - / arctg ~dx = ~ . x I0 -

000

X

x

1

In (1

X2)) 12

7r

7r

= In 2

- 2 ( - arctg -

- -

+ -

= - -

2 . - + In 2

222

4024

(HHTerpaJI /

arctg ~ dx 6bIJI Haii.n;eH HHTerpHpoBaHHeM no qacTHM).

•

B'bt"lUc.!tum'b

uHmezpa.!t'b/,:

3.1.21.

//(4 -

x 2 - y2) dxdy, r.n;e 06JIacTb D OrpaHHqeHa JIHHHHMH x = 0,

D

= 1,5.

Y = 0, x = 1, y

3.1.22.

//(3 -x - y) dxdy, r.n;e D -

Kpyr x 2 + y2 :::;; 1.

D

1)2 + (y - 1)2 :::;; 1.

3.1.23.

/ /

xy dxdy, r.n;e D -

Kpyr (x -

D

Jx 2 + y2 dxdy, r.n;e D -

Kpyr x 2 + y2 -

3.1.24.

/ /

2ax :::;; O.

3.1.25.

D

IhMeHHTb nopH.n;oK HHTerpHpoBaHHH B nOBTopHOM HHTerpaJIe

6

3+v'12+4x-x2

/

dx

/

!(x, y) dy.

-2 3-v'12+4x-x2

a YqHTbIBM npe.n;eJIbI HHTerpHpOBaHHH, npe.n;CTaBHM 06JIacTb D B BH.n;e CHCTeMbI HepaBeHCTB

{ -2:::;; x:::;; 6,
3 -	y''"I-;:O-2-:-+-47""X-_-X"2 :::;; y :::;; 3 + v'12 + 4x - x 2.
rpacpHKH <PYHKU;Hii	YI	= 3 - v'12 + 4x - x 2 H	Y2 = 3 + v'12 + 4x - x 2
IIpe.n;CTaBJIHIOT c060ii COOTBeTCTBeHHO HIDKHIOIO H BepXHIOIO nOJIYOKpY:>KHo-
CTH OKPY:>KHOCTH (y -		3)2 = 12 + 4x - x 2, HJIH (x - 2)2 + (y - 3)2 = 16.
TaKHM 06pa30M, 06JIacTb HHTerpHpOBaHHH D -			Kpyr pa,!l;Hyca 4 C u;eHTpoM
B TOqKe (2,3) (pHC.	16). 3a,rr.a)J;HM 9TOT Kpyr .n;pyroii CHCTeMoii HepaBeHCTB.

ECJIH cnpoeKTHpOBaTb ero Ha OCb Oy, TO nOJIyqHM oTpe30K [-1,7]' oTKy.n;a HMeeM nepBoe HepaBeHcTBo -1 :::;; y :::;; 7. Bblpa3HB .n;aJIee x H3 ypaBHeHHH OKPY:>KHOCTH, nOJIyqHM COOTBeTCTBeHHO ypaBHeHHH JIeBoii H npaBoii nOJIyOKPY:>KHOCTeii Xl = 2 - J16 - (y - 3)2 H X2 = 2 + J16 - (y - 3)2. Tenepb

135

06JIacTb D MO:lKHO 3aIUI:CaTb TaK:
D'	{-I ~ y ~ 7,
.	2 - Jr"16°-----;-(y------,-,3)"""""2 ~ X ~ 2 + J 16 - (y - 3) 2 •

TaKHM 06pa30M, nOCJIe 3aMeHbI nOpH,n;Ka HHTerpHpOBaHHH HCXO,n;Hblit nOBTOpHblit HHTerpaJI MO:lKHO 3anHCaTb B BH,n;e

7	2+v'16-(y-3)2			•
!	dy	!	f(x, y) dx.
-1	2-v'16-(y-3)2
			y
				c
				x
				B
Puc. 16			Puc.	17
			'1/2	y2/2
3.1.26. IhMeHHTb nopH,n;oK HHTerpHpoBaHHH ! dy				! f(x, y) dx.
			-'1/2	y2_1
a IIpH pa360pe 9Toro npHMepa HCnOJIb3yeM ,n;pyroit rro,n;xo,n;. 06JIaCTb HH-
TerpHpOBaHHH D 3a,n;aeTCH CHcTeMoit HepaBeHCTB
	v'2 ~ y ~ v'2,
	{ 2		y2
	y	-1~x~2'

reOMeTpH'leCKH9TO 03Ha'laeTCJIe,n;yIOuree: Ka:lK,n;M ropH30HTaJIbHaH rrpHMM,

rrpoxo~urM 'lepe3TO'lKHOTpe3Ka [-v'2, v'2]	OCH Oy, nepeceKaeT CHa'laJIa
(npH ,n;BH:lKeHHH CJIeBa HarrpaBO) rrapa60JIy X	= y2 - 1 (Ha30BeM ee JIHHHeit
2

Bxo,n;a B D), a 3aTeM rrapa60JIY X = Y2 (Ha30BeM ee JIHHHeit Bblxo,n;a H3 D) -

CM. pHC. 17.

IIpH nepeMeHe rropH,n;Ka HHTerpHpOBaHHH Hy:lKHO cnpoeKTHpOBaTb 06JIacTb HHTerpHpOBaHHH D Ha ,n;pyryIO OCb (OCb Ox) H o6HapY:lKHTb JIHHHH Bxo,n;a H Bblxo,n;a rrpH ,n;BH:lKeHHH CHH3Y BBepx B,n;OJIb BepTHKaJIbHbIX rrpHMblx.

136

2

IIapa60JIbI x = Y2 H x = y2 - 1 IIepeceKalOTCH B TOqKax B(I, -J2) H

y2

C(I, J2) (.n.eikTBHTeJIbHO, IIpHpaBHHBM ypaBHeHHH IIapa60JI, HMeeM 2 =

== y2 _ 1 ¢} y2 = 2 ¢} Y = ±J2). TaKHM 06Pa30M, IIpoeKIIHH 06JIaCTH D Ha OCb Ox - OTpe30K [-1,1]. 113 pHcYHKa BH.n.HO, qTO Ha yqacTKe x E [-1,0]

TOqKH Bxo.n.a H Bblxo.n.a pacIIOJIO)KeHbI Ha BeTBHX o.n.Hofi IIapa60JIbI, a Ha yqaCTKe x E [0, 1] - Ha BeTBHX pa3HbIX IIapa60JI. CHaqaJIa oIIpe.n.eJIHM BeTBH

3THX IIapa6oJI, peIIIM OTHOCHTeJIbHO y ypaBHeHHH x	y2
	= y2 - 1 H X = 2	Ha
COOTBeTCTBYIOIIIHX yqacTKax. IIoJIyqaeM: y = ±Vx+1 H Y = ±$x, x		~ o.
IIepBoe paBeHCTBO COOTBeTcTByeT .n.yraM AC (3HaK	«IIJIIOC») H AB (3HaK
«MHHYC»), BTopoe - .n.yraM OC H OB (pHC. 17). TeM caMbIM, 06JIaCTb D

pa36HBaeTcH Ha TpH OT.n.eJIbHhle 06JIacTH D 1 , D2 H D3 , T. e. D = DI U D2 U D3 , r,ne

D . {-I ~ x ~ 0,	D . {o ~ x ~ 1,
I· -Vx+1 ~ Y ~ Vx+1;		2· -Vx+1 ~ Y ~ -$x;
	D 3 ' {o ~ x ~ 1,
	. $x ~ y ~ Vx+1.
lfCXO.n.Hblfi HHTerpaJI HaIIHIIIeM B BH.n.e .n.BofiHoro
v'2	y2/2	II !(x,y) dxdy,
I dy	I !(x, y) dx =
-v'2	y2_1	D

H IIpHMeHHH CBoficTBO a,II.,LLHTHBHOCTH .n.BofiHoro HHTerpaJIa, 3aIIHIIIeM OTBeT

II !(x, y) dxdy =			o	Vx+I
			I	dx	I	!(x, y) dy+
D			-I	-Vx+I
				1		-v'2X				1	Vx+I
			+ I dx			I	!(x, y) dy + I dx				I !(x,y)dy. •
				o	-Vx+I					0	v'2X
H3.MeHUmb nopSliJo'IC UHmeepupoeaHUS!:
	3	3-y								1	2-y
3.1.27.	I dy	I !(x, y) dx.					3.1.28.			I dy	I !(x, y) dx.
	o	0	!(x, y) dy + I3 dx I3							0	y
3.1.29.	Io dx	I3						!(x, y) dy.
	-3	-x				0	x
		JI-y		2			v'2/2 JI-y 2
3.1.30.			I	!(x,y)dx +			I	dy	I	!(x,y) dx.
		-y					o		y

137

3.1.31. BblqHCJIHTb HHTerp8.JIbHOe cpe,n:Hee 3HaqeHHe <PYHKIJ;HH Z			= 12-:.:
	- 2x -	3y B 06JIaCTH D, OrpaHHqeHHoit npHMbIMH 12 - 2x -	3y = 0,
a	x = 0, y = O.
	06JIacTb D -	TpeyrOJIbHHK OAB, r,n:e 0(0,0), A(6, 0), B(O, 4) -	pHC. 18.

y

x

Puc. 18

ITo onpe,n:eJIeHHIO HHTerp8.JIbHOe cpe,n:Hee 3HaqeHHe <PYHKIJ;HH z(x,y) B

06JIaCTH D paBHO	~ II z(x,y)dxdy, r,n:e S - nJIOrn;a,n:b 06JIacTH D (CBOit-
CTBO 3).	D

ITJIOrn;a,n:b S BblqHCJIHeM no <p0pMYJIe nJIOrn;a,n:H npHMoyrOJIbHOrO TpeyrOJIbHHKa: S = !IOAI . lOBI = 12. OCTaeTcH BhlqHCJIHTb HHTerp8.JI no o62a- CTH D, KOTOPYIO MO:lKHO 3a,n:aTb HepaBeHCTBaMH 0 ~ x ~ 6, 0 ~ y ~ 4 - aX. lIMeeM

			2
		6 4- 3 x
11(12-2x-3y)dxdy= I		dx	I (12-2x-3y)dy=
D	0		0
	6	4- ~x	6	12 -	2x
	Idx(12Y-2xy-~y2)lo 3		= 1[(12~2x)y-~y2]lo 3		dx=
	o		0

6 1	2	1 (12 - 2x)3	6
= 16(12-2x) dX=-6 3.2			10=48.
o
TaKHM o6pa30M, HCKOMoe HHTerp8.JIbHOe cpe,n:Hee paBHO 1~, T. e. 4.			•

B'bt"tuc.n,um'b U'Hmeepa.n,'b'H'bte cpea'Hue 3'Ha"te'H'UJI aa'H'H'btx tjjy'Hr.;v,u1J, 6 yr.;a3a'H- 'H'btX o6.n,ac'1nSlx:

3.1.32.!(x,y) = 2x + y, D - TpeyrOJIbHHK OAB C BepIIIMHaMH 0(0,0),

A(0,3), B(3,0).

3.1.33. !(x, y) = x + 6y, D - TpeyrOJIbHHK, OrpaHHqeHHblit npHMbIMH

y = x, y = 5x, x = 1.

138

3.1.34.	f (x, y) = JR2 -				X2 -	y2, D -		Kpyr X2 + y2 ~ R2.
3.1.35.	f(x,y) = x 3 y2, D -					Kpyr X 2 + y2 ~ R2
AononHIIITenbHble 3aWlHIIIR
3.1.36.	Ii		. x+y+lO
			sm	2	2	dxdy.
	x2+y2::;;4		X	+y +5
3.1.37.	II	xy(x + y) dxdy.
	0::;;x::;;3
	0::;; y::;; 3
3.1.38.	II (x 2 + y2 -				2 Jx 2 + y2) dxdy.
	0::;;x::;;2	II
3.1.39.	0::;;y::;;2				(x2 + y2 -		4x -	4y + 10) dxdy.
	(x-l)2+4(y-2)2::;;4
OnpeOe.l/,Umb 3'1ta'l> oa'lt'lt'btx u'ltmezpa.l/,oe:									II Vl- x 2 - y2 dxdy.
3.1.40.	II		In(x 2 + y2) dxdy.				3.1.41.
	Ixl+lyl::;;l								x2+y2::;;4

3.1.42.II arcsin(x + y) dxdy.

O::;;x::;;l -l::;;y::;;l-x

,4eo1i.'lto1i. U'ltmeZpM II f(x, y) dxdy no 3aOa'lt'lto1i. o6.1/,acmu npeocmaeumb e

D

6uoe noemop'ltozo oey.MSI cnoco6aMu. COe.l/,amb "lepmeJIC o6.1/,acmu u'ltmezpup06a'lt'USl.:

3.1.43.D orpaHFPleHa JIHHHHMH y = 0, x = 5, y = x.

3.1.44.D - TpeyrOJIbHHK C BepWHHaMH B TOqKax A(-1, -1), B (1, 3),

C(2, -4).

3.1.45.D - IIapaJIJIeJIOrpaMM ABCD C BepWHHaMH A( -3,1), B(2, 1),

C(6, 4), D(I,4).

3.1.46.D - Kpyr (x - 2)2 + (y - 3)2 ~ 4.

3.1.47.D OrpaHHqeHa JIHHHHMH y = x2, X = y2.

3.1.48.	D OrpaHHqeHa JIHHHHMH y = x3 , X + y = 10, x -				y = 4, y = o.
	2	2x		y2x-x2
3.1.49.	I dx I fdy.		3.1.50.	I	fdy.
	o	x		2-x

139