Сборник задач по высшей математике 2 том

4.1.70.	f(x,y) = ye- X , L -		yqacTOK KPHBOt;i
	x = In(l + t 2 ), y = 2arctgt -			t + 3,
	3a,Ll;aHHot;i IIapaMeTpHqeCKH, Me)K,LJ;y TOqKaMH, COOTBeTCTBYIOm:HMH
	t=OHt=1.	,LJ;yra IIapa60JIbI y = 2x, JIe)Kam:aH M€)K,LJ;y TOq-
4.1. 71.	f (x, y) = ~, L -
4.1. 72.	KaMH (1, V2) H (2,2).
	f(x, y) = y3, L -	apKa IJ;HKJIOH,lJ.bI
	x = a(t - sint),		y = a(l- cost),	0 ~ t ~ 211".

§ 2. KPLJlBOflLJIHEUiHblUi LJlHTErPAfl BTOPOrO POAA OnpeAeneHMe KpMBonMHe~Horo MHTerpana BToporo POAa

~ IIycTb L = AB - ma,n.KaH KpHBaH, a P(x, y) - HeKOTOpaH <PYHKIJ;HH, onpe- ,!I;eJIeHHaH B TO'lKaxKpHBOil: L. Pa306heM KPHBYIO L Ha n npOH3BOJIbHhIX 'IacTeil: TO'lKaMH A = Mo, MI, M2, ... , Mn = B. ,I1;aJIee Ha KruK,!I;Oil: 1'13 nOJIY'IeHHhIX

,!I;yr ~ BhI6epeM npOH3BOJIbHYIO TO'lKYMi(xi, ih), nOCJIe 'IeroCOCTaBHM npo-

H3Be,!l;eHHe P(Xi,Yi) ~Xi 3Ha'leHHH<PYHKIJ;HH P(x,y) B TO'lKe Mi Ha npOeKIJ;HIO

~Xi = Xi+l - Xi 3TOil: ,!I;yrH Ha OCb OX. CKJIa,Il.hIBaH Bce TaKHe npOH3Be,!l;eHHH, no-

JIY'IHMCYMMY

n

Sn,,, = L P(Xi, Yi)~Xi,

i=O

KOTOPaH Ha3hIBaeTCH tmmeZpaJI'bHot1. CYM.M.ot1. emopozo poiJa ,!I;J1H <PYHKIJ;HH P(x, y) no KOOp,!l;HHaTe x.

IIycTb Tenepb d - HaH60JIbllaH 1'13,!I;J1HH,!I;yr M-:'Mi. ECJIH <PYHKIJ;HH P(x, y)

HenpephIBHa B TO'lKaxKPHBOil: L, TO npH d ~ 0 CYIIJ;eCTByeT npe,!l;eJI HHTerpaJIbHhIX CYMM Sn,,, , He 3aBHCHIIJ;HiI: OT cnoc06a pa36HeHHH KPHBOil: L Ha 'IacTH1'1 BhI60pa TO'leK Mi. 9TOT npe,!l;eJI Ha3hIBaeTCH .,.pueOJltmet1.H'b/,M. UHmeZpaJIOM. emopozo poiJa no KOOp,!l;HHaTe X 1'1 0603Ha'iaeTCH

j P(x,y)dx.

L

AHaJIOrH'IHOonpe,!l;eJIHeTCH 7CpUeOJlUHet1.H'b/,t1. uHmeZpaJi emopozo poiJa no KOOP- ,!I;HHaTe y, KOTOPhIiI: 0603Ha'iaeTCH

j Q(x,y)dy,

L

r,!l;e Q(x, y) - HenpephIBHaH <PYHKIJ;HH.

~CYMMa KPHBOJIHHeil:HhIx HHTerpaJIOB

jP(X,y)dX 1'1 j Q(x,y)dy

L L

200

Ha3bIBaeTCH nO.ll'lt'btM 'lCPU60.llU'lteit'lt'bt.M u'ltmeepa.llO.M 6mopoeo poiJa H 0603Ha'laeTCH

j P(x, y) dx + Q(x, y) dy.

L

KPHBOJIHHeitHble HHTerpaJIbI BToporo pO)l;a Ha3b1BaIOTCH TaIOKe KPHBOJIHHeit-

HbIMH HHTerpaJIaMH rro KOOp)l;HHaTaM.

KPHBOJIHHeitHbIit HHTerpaJI BToporo pO)l;a 06JIa,!l;aeT TeMH lKe CBoitcTBaMH, 'ITO

H orrpe)l;eJIeHHblit HHTerpaJI. B '1acTHOCTH,

j P(x, y) dx + Q(x, y) dy = -	j P(x, y) dx + Q(x, y) dy,
BA	AB

T. e. KPHBOJIHHeitHbIit HHTerpaJI BTOpOro pO)l;a MeHHeT 3HaK rrpH H3MeHeHHH HarrpaBJIeHHH HHTerpHpOBaHHH.

Bbl'"lMClleHMeKPMBOIlMHeiiiHblX MHTerpallOB BTOpOrO pOAa

IIpe)l;rroJIOlKHM, 'ITOKpHBaH L 3a,!l;aHa B HBHOM BH)l;e HerrpepbIBHO )l;H<p<pepeHJJ:HpyeMoit <pYHKU;Heit y = y(x), X E [a,b]. Tor)l;a

	b
j	P(x, y) dx + Q(x, y) dy = j[P(x, y(x)) + Q(x, y(x))y'(x)] dx.
L	a

ECJIH L 3a)l;aeTCH rrapaMeTpH'IeCKHMH<PYHKU;HHMH X = x(t), y = y(t), t E [a, ,a],

TO
	(3
j	P(x, y) dx + Q(x, y) dy = j[P(x(t), y(t))x'(t) + Q(x(t), y(t))y'(t)] dt.
L	'"

:::ho paBeHCTBO MOlKHO pacrrpocTpaHHTb H Ha rrpocTpaifcTBeHHbIit CJIY'Iait(aprYMeHTbI (X, y, z) <PYHKU;Hit P, Q, R )l;JlH KpaTKOCTH orrycKaeM):

	(3
j	Pdx + Qdy + Rdz = j(P' x'(t) + Q. y'(t) + R· z'(t)) dt,
L	'"

r)l;e (X, y, z), X = x(t), y = y(t), z = z(t) - rrapaMeTpH'IeCKHeypaBHeHHH KPHBOit L.

npMIlO)l(eHMJI KPMBOIlMHeiiiHOrO MHTerpalla BTOpOrO POWI

HHTerpan

jPdx+Qdy

L

MOlKHO rrpe)l;CTaBHTb B BH)l;e CKaJIHPHOro rrpOH3Be)l;eHHH BeKTopOB F = Pi + Qj H

ds = i dx +j dy:

jPdx+Qdy= jF(x,y).ds.

L L

201

H (x 1, yt)

B TaKOM CJIY'Iae

IF.ds

L

BhIpaJKaeT pa60TY nepeMeHHoi!: CHJIbI F = Pi+Qj npH nepeMem;eHHH MaTepHaJIbHoi!: TO'lKHM = M(x, y) B,1I;OJIb KpHBOi!: L = AB OT TO'lKHA,1I;o TO'lKHB.

IIpH A = B KpHBaH L 3aMKHYTa, a COOTBeTCTBYJOm;Hi!: KPHBOJIHHei!:Hbli!: HHTerpaJI no 3aMKHYToi!: KPHBOi!: 0603Ha'laeTCHTaK:

f Pdx+Qdy.

B 9TOM CJIY'IaeHanpaBJIeHHe 06xo,1l;a KOHTypa HHOr,1l;a nOHCHHeTCH CTPeJIKOi!: Ha KpYlKKe, pacnOJIOlKeHHOM Ha 3HaKe HHTerpaJIa.

IIpe,1l;noJIOlKHM, 'ITOB nJIOCKOCTH Oxy HMeeTCH O,1l;HOCBH3HaH 06JIaCTb D (9TO 3Ha'lHT,'ITOB Hei!: HeT «)1;hIp») , OrpaHH'IeHHaHKpHBOi!: L = aD (aD - 0603Ha'leHHe I'paHHD;bI06JIaCTH D), a B 06JIaCTH D H Ha ee rpaHHD;e aD <PYHKD;HH P(x, y) H Q(x, y)

HenpepblBHbI BMeCTe co CBOHMH '1acTHblMHnpOH3BO,1l;HblMH.

TeopeMa 4.1. nYCTb A M B - npOM3BOJlbHbie TO'iKM06JlaCTM D. AmB M AnB-

ABa npOM3BOJlbHbiX nYTM (maAKMe KpMBble). COeAMHftlOl.I.IMe 3TM TO'iKM (pMC. 45).

TorAa CJleAYlOl.I.IMe YCJlOBMft paBHOCMJlbHbl:

1.	aQ	ap
	ax =	ay (YCJlOBMe rpMHa).
2.	I Pdx + Qdy = I Pdx + Qdy (KpMBOJlMHeIilHbllii MHTerpaJl He 3aBMCMT
	AmB		AnB
OT nYTM MHTerpMpOBaHMft).
3.	I	Pdx+Qdy = 0 (MHTerpaJl no Jl1060MY 3aMKHYTOMY nYTM paBeH HYJlIO).
	AnBmA		= dU (Bblpa>KeHMe Pdx + Qdy npeACTaBJlfteT co6olil nOJlHbl1ii
4.	Pdx	+ Qdy
AMcPcPepeHLlMaJl			HeKoTopolii cPYHKLlMM U = U(x, y)).

Puc. 45

B CJIY'IaeBbinOJIHeHHH JIJ060ro H3 paaHOCHJIbHblX YCJIOBHi!: npe,1l;bI)1;ym;ei!: TeopeMbi KPHBOJIHHei!:Hbli!: HHTerpaJI no JIJ060i!: KPHBOi!:, COe,1l;HHHJOm;ei!: TO'lKH(xo, YO)

H3 o6JIacTH D, MOlKHO BbI'IHCJIHTb npH nOMOm;H <P0PMYJIbi HbJOToHa-

202

JIeii6HHIJ;a
(Xjl,YI)	I(XI ,ytl
Pdx+Qdy = U(x,y)		= U(Xl,y!) -	U(xo, YO),
(xO,YO)	(XO,YO)
me U(x, y) - HeKOTOpruI rrepBoo6pa3HruI ,!VIH P dx + Q dy.
C ,1I;Pyroii CTOPOHbI,	rrepBoo6pa3HaH	U(x,y) BblpaJKeHHH Pdx + Qdy MOlKeT
6bITb Haii,1l;eHa rrpH rrOMoru;H KPHBOJIHHeiiHOro HHTerpaJIa
	(x,y)
	U(x,y) = j	Pdx + Qdy.
	(XO,YO)
B 9THX lKe YCJIOBHHX Ha <PYHKIJ;HH P(x,y) H Q(x,y), a			TaKlKe Ha 06JIaCTb D,

HMeeT MeCTO ifjopMyJla rpU'Ha, rr03BOJIHIOru;ruI CBeCTH KPHBOJIHHeil:HbliI: HHTerpaJI rro 3aMKHYTOMY KOHTYPY K ,1I;BOil:HOMY HHTerpaJIY

f	P dx + Q dy = jj (~~ - ~~) dxdy.
aD	D

3!J:eCb rrpe,1l;rrOJIaraeTCH, 'ITOo6xO!J: rpaHHIJ;b1 aD 06JIacTH DB KPHBOJIHHeil:HoM HHTerpaJIe

fPdx+Qdy aD

COBepIIIaeTCH B rrOJIOlKHTeJIbHOM HarrpaBJIeHHH, T. e. rrpH TaKOM o6xo,1l;e rpaHHIJ;b1 06JIaCTb D OCTaeTCH CJIeBa; ,!VIH O,1l;HOCBH3HOil: 06JIacTH 9TO HarrpaBJIeHHe COBrra,!l;aeT C HarrpaBJIeHHeM rrpOTHB '1acOBOil:CTPeJIKH.

3aMeTHM, 'ITOrrJIoru;a,!I;b S = S(D) 06JIaCTH D MOlKeT 6b1Tb BbI'IHCJIeHarrpH nOMOru;H KPHBOJIHHeil:Horo HHTerpaJIa BToporo pO,1l;a:

S = l f x dy - y dx aD

(2lTa <p0pMYJIa rrOJIY'IaeTCHH3 <P0PMYJIbI rpHHa C P = -ly, Q = ly)·

4.2.1. ,naHbI <PYHKIIHH P(x, y) = 8x + 4y + 2, Q(x, y) = 8y + 2 H TO'lKH

A(3, 6), B(3,0), C(0,6). BbI'lHCJIHTbKPHBOJIHHeitHblit HHTerpan

j(8X + 4y + 2) dx + (8y + 2) dy,

L

r,ll,e:

1)L - OTpe30K OA;

2)L - JIOMaHaH OBA;

3)L - JIOMaHaH OCA;

4)L - rrapa6oJIa, CHMMeTpH'lHaHOTHOCHTeJIbHO OCH Oy H rrpoxo-

)J.fIIIIaH 'lepe3TO'lKH0 H A;

5) rrpOBepHTb BbIIIOJIHHMOCTb YCJIOBHH rpHHa.

203

Q IIYTH HHTerpHpOBaHHfI, COOTBeTCTBYIOID:He II. II. 1)-4), H306pax<eHbI Ha'

pHc.46.

y

(7

o

6				A
--------------				:
				'II
				,","
			,	,
			",
			,	,
		,		I
	,,,I		,,,I
	,		,
,,,,		,,,,
,		,
,,,	,,,
,,,,,",,,,,"
" ... '~				3

B x

Puc. 46

1) OTpe30K OA MO)l{eT 6bITb 3aIIHcau B BH,n:e: y = 2x, x E [0,3]. Tor,n:a

dy = 2dx H

3

!Pdx+Qdy= ![(8x+4.2X+2)dx+(8.2X+2).2dx] =

OA	0
	3	3

=P48X + 6) dx = (24x2 + 6x)\o = 234. o

2)lIcIIoJIb3yeM cBoiicTBO a,II,IJ;HTHBHOCTH, BblqHCJIfifi OT,n:eJIbHO HHTerpaJI

IIO oTpe3KaM OB H BA. Tor,n:a:

a) OB: 3,n:eCb y = 0, 0 ~ x ~ 3, T. e. dy = 0, oTKy,n:a

3 3

!(8x + 4y + 2) dx + (8y + 2) dy = P8x + 2) dx = (4x2 + 2x)\o = 42.

DB	0
6) BA: x = 3, 0 ~ y ~ 6, T. e. dx = 0, H
	6	6

!(8x + 4y + 2) dx + (8y + 2) dy = P8y + 2) dy = (4y2 + 2y)\o = 156.

,BA

0

TaKHM 06Pa30M,

! (8x + 4y + 2) dx + (8y + 2) dy = 42 + 156 = 198.

DBA

3) 9TOT HHTerpaJI BblqHCJIHM auaJIOrHqHO IIpe,II.bI,n:yw;eMy.

204

a) OC: x = 0,			(T.e. dx = 0), 0 ~ y ~ 6, oTKY,ll,a
						6
	j (8x + 4y + 2) dx + (8y + 2) dy = j(8Y + 2) dy = 156.
	oc					0
6) CA: 0 ~ x ~ 3, y = 6, dy = 0, CJIe,ll,OBaTeJIbHO,
						3
	j(8X + 4y + 2) dx + (8y + 2) dy = j(8x + 26) dx = 114.
	CA					0
OKOH'IaTeJIbHO
	j	(8x + 4y + 2) dx + (8y + 2) dy = 114 + 156 = 270.
	OCA
4) IIo,ll,cTaBHB KOOp,ll,HHaTbI TO'IKHA(3j 6) B paoeHcTBo y = ax2 Haii,n,eM
				2x2				4
ypaBHeHHe ,ll,aHHoi!: napa60JIbI y = T. IIpH STOM 0 ~ X ~ 3								H dy = aX dx,
						C"'.
OTKY,ll,a (nYTb OA no napa60JIe 0603Ha'IHM OA)
j	(8x + 4y + 2) dx + (8y + 2) dy =
Q	= j[(8X+ 8t +2) dx+ e~2 +2) ~XdX] =
		o
	3	~x2 + 32x + 2) dx -						3
= j(64 x 3 +					(16 x 4 + ~x3 + 16 x 2 + 2x) 1- 222
o	9	3	3	-	9	9	3	0 - ·
5) l:IMeeM
	~~ = tu (8x + 4y + 2) = 4,					~~ = tx (8y + 2) = 0,
T. e. YCJIOBHe rpHHa He BbIIIOJIHHeTCH. 9TOT cPaKT, a							TaJOKe BbI'IHCJIeHHHB

nYHKTax 1)-4) STOi!: 3a,n:a'IHIIOKa3b1BaIOT, 'ITO,ll,aHHbIi!: KPHBOJIHHei!:HbIi!: HH-

Terpan BToporo pO,ll,a 3aBHCHT OT nYTH HHTerpHpOBaHHH.		•
4.2.2.	,r:t;aHbI cPYHKIIHH P(x, y) = y + 3, Q(x, y) = 8x + 7y + 6 H TO'IKH

A(9,4), B(9,0), C(0,4). BbI'IHCJIHTbKPHBOJIHHei!:HbIi!: HHTerpan

j(y + 3) dx + (8x + 7y + 6) dy,

L

r,ll,e:

1)L - OTpe30K OAj

2)L - JIOMaHaH OBA;

3)L - JIOMaHaH OCA;

4)L - napa60JIa, COe,ll,HHHIOIIIaH TO'IKH0(0,0) H A(9,4) H CHM-

MeTpH'IHaHOTHOCHTeJIbHO OCH Oy.

5) IIpoBepHTb BbIIIOJIHeHHe YCJIOBHH rpHHa.

205

OT n ,0:0 0),

4.2.3.

1(4y + 4) dx + (3x + 3y + 4) dy

L

IIO pa3HbIM "YT5IM, coe,o:HH5IIOIIIHM TO'IKH0(0,0), A(2,6), B(2,0) C(0,6):

1)L = OA;

2)L = OCA;

3)L = OBA;

4)L - .n:yra 6A IIapa60JIbI y = ~x 2 •

4.2.4.BbI'IHCJIHTbHHTerpan

12XY dx - x 2 dy,

L

B351TbIit B,o:OJIb pa3JIH'IHbIXIIYTeit, Coe,o:HH5IIOIIIHX TO'IKH0(0,0)

A(2, 1),		B(2, 0),	C(0,1):
1)	L -	oTpe30K OA;
2)	L -	IIapa60JIa C OCbIO CHMMeTpHH Oy, IIpOXO.D:»IIIM 'Iepe30

H A;

3)L - IIapa60JIa, IIpOXO.D:»IIIM 'Iepe30 HAc OCbIO cHMMeTpHIf

Ox;

4)L - JIOMaHM OBA;

5)L - JIOMaHa51 OCA.

4.2.5.BbI'IHCJIHTbHHTerpan

	1 y2 dx + x 2 dy,
	L	2	y2
	r,o:e L - BepXH5I5I IIOJIOBHHa 9JIJIHIICa	X 2	+ 2 = 1, IIp06eraeMM
	IIO XO.n:Y 'IacOBOitCTpeJIKH.	a	b
Q	BOCIIOJIb3yeMc5I IIapaMeTpH'IeCKHMHYPaBHeHH5IMH 9JIJIHIIca: x = a cos t,
y	= bsint, t E [O,n], T.e. dx = -asintdt, dy	=	bcostdt. TIo,o:cTaBJI»» B

HHTerpan H y'IHTbIBMHaIIpaBJIeHHe 06xo,o:a (oTKy,o:a CJIe.n:yeT, 'ITOt MeH5IeTC5I

IIOJIY'IaeM

	o
1 y2 dx + x 2 dy = 1(_b2 sin2 t . a sin t + a2 cos2 t . b cos t) dt =
L	7r
	7r	7r
	= 1ab2 sin2 t . sin t dt -	1a2b cos2 t . cos t dt =
	o	0
	= -ab2 1(1 -cos2 t) d( cos t) - ab2 1(1 -sin2 t) d(sin t) =
	7r	7r
	o	0	I:= ~ab2. •
	= -ab2 (cos t - cOf t)	I:_a2b (sin t _ Si~3t)

206

OT 0 ,0:0 21r.

4.2.6.		j	x 2 dy - y2 dx
					55'
		L		X3 + y3
	r,o:e L -	.n:yra KPHBOil: x = R COS3 t, Y = R sin3 t, np06eraeMa» OT
	TOqKH A(R, 0) K B(O, R).
4.2.7.				j	xydx,
				L
	r,o:e L -.n:yra CHHyCOH,lI,bI y = sin x OT TOqKH (0,0) ,0:0 TOqKH (1r,0).
4.2.8.	BblqHCJIHTb			j	xdy,
				L
	r,o:e L -	OTpe30K np»Moil: ~ + t = 1 OT TOqKH A(a,O) ,0:0 TOqKH
	B(O, b).
4.2.9.
		j(x2 _ y2) dx + (x2 + y2) dy
		L
		2		y2
	B,o:OJIb 9JIJIHnCa x2 +			2	= 1, np06eraeMoro B nOJIO:lKHTeJIbHOM
		a		b

HanpaBJIeHHH (npoTHB qaCoBoil: CTpeJIKH).

4.2.10.BblqHCJIHTb

j yzdx + xzdy + xydz

L

no .n:yre BHHTOBOil: JIHHHH x = a cos t, y = a sin t, z = bt npH H3Me- , HeHHH t

Q CHaqaJIa Hail:,n:eM ,o:H<p<pepeHIJ;HaJIbI nepeMeHHbIX: dx = -a sin t dt, dy =

=a cos t dt, dz = bdt. Bblpa3HM no,o:bIHTerpaJIbHOe Bblpa:lKeHHe qepe3 t, CBO,ll;» HCXO,o:Hblil: HHTerpaJI K onpe,n:eJIeHHoMY:

	211"
j yzdx + xzdy + xydz =	j (-a2btsin2 t + a2btcos2 t + ba2 sint cos t) dt =
L	0
	211"
= a2b j (t cos 2t + si;2t) dt =
	o
=a2b(t. Si;2tl:1I" _~ lSin2tdt- cO~2tl:1I") =0. •
	o
4.2.11.	j xdy - ydx,
	L
r,o:e JIHHH» L -	3a,n:aHa ypaBHeHH»MH x = 2V5 cos3 t, Y = 4V5 sin3t,
tE [0, 21r].

207

4.2.12.	BblqHCJIHTb	+ y2)3 dx
	j(x2
	L
	B,n:OJIb OKP)')KHOCTHx 2 + y2 = 5, rrpo6eraeMoii B rrOJIOlKHTeJIbHOM
	HanpaBJIeHHH.
4.2.13.	BblqHCJIHTb
	j(X2 - 2xy2 + 3) dx + (y2 -		2x2y + 3) dy,
	L
	r,n:e L - A)'rarrapa60JIbI y = ax2 ,		coe,n:HH5IIOIII.eii TOqKH 0(0,0) H
	A(2,8).

4.2.14. BblqHCJIHTb

j ydx + zdy + xdz,

L

r,n:e L	- BHTOK BHHTOBOii JIHHHH x	= a cos t, y = a sin t, z = bt,
o~ t	~ 211', rrpo6eraeMblii B HarrpaBJIeHHH y6bIBaHH5I rrapaMeTpa.
4.2.15. IIoKa3aTb, qTO HHTerpaJI
	(10,10)
	j (x + y) dx + (x -	y) dy
	(0,0)

He 3aBHCHT OT rryTH HHTerpHpOBaHH5I, coe,n:HH5IIOIII.ero TOqKH (0,0) H (10,10), H BblqHCJIHTb ero.

Q IIpoBepHM YCJIOBHe rpHHa. IIoJIo)KHM P = X + y, Q = x - y. Tor,n:a

oQ _ oP- 1

ox - oy - ,

H, 3HaqHT, ,n:aHHblii HHTerpaJI ,n:eiiCTBHTeJIbHO He 3aBHCHT OT nyTH HHTerpHPOBaHH5I. ,1J;JI5I BblqHCJIeHH5I ,n:aHHOrO HHTerpaJIa B KaqeCTBe rryTH HHTerpH-

POBaHH5I B03bMeM rrpocTeiimHii, T. e. OTpe30K, COe,n:HH5IIOIII.Hii TOqKH 0(0,0) H B(lO, 10). OTpe30K OB MO)KHO 3a.u,aTb TaK: y = x, x E [0,10]. IIpH 9TOM dy = dx, H HHTerpaJI JIerKO CBO,n:HTC5I K orrpe,n:eJIeHHOMY HHTerpaJIY

(10,10)	10	10
j (x + y) dx + (x -	y) dy = j (x + x) dx = x210 = 100.		•
(0,0)	°

IIpoaepumb, ",mo oaHHwe tcpuao.lluHeilHWe UHmeapa.llW He 3aaUCJlm om nymu UHmeapUpOaaH'UJI U aW"'UC.IlUmbux:

(1,1)

4.2.16.j (3x2 - 3y) dx + (3y2 - 3x) dy.

(0,0)

(2,0)

4.2.17.j (3x 2 + 6xy2) dx + (6x2y + 4y3) dy.

(1,1)

208

	2,3	3Xy2 + 2) dx -
4.2.18.	/ (X3 -		(3x2y -	y2) dy.
	(0,0)
4.2.19.	BbPIHCJIHTb KPHBOJIHHeil:Hblil: HHTerpa.rr
		/(x + 1) dx + xyz dy + y2 z dz,
		L
	r,ll,e L -	OTpe30K, COe,ll,HHHIOIIIHil: TO'lKY C(2, 3, -1) C TO'lKOil:
	D(3, -2,0).
a CocTaBHM rrapaMeTpH'IecKHeypaBHeHHH OTpe3Ka CD, HCrrOJIb3YH ypaB-
HeHHH rrpHMoil:, rrpOXO,ll,HIIIeil: 'Iepe3,ll,Be TO'lKH:
		x-2	y-3	z+1
		- 1 - = --=5		-1-·
OTCIO,ll,a x	= 2 + t, y = 3 - 5t, z = -1 +t, t			E [0, 1]. ,r:t;a.rree, HaxO,ll,HM dx = dt,

dy = -5 dt, dz = dt, rrO,ll,CTaBJIHeM Bce H~Hble BblpaJKeHHH B ,ll,aHHbIil: HHTerpa.rr, 0603Ha'leHHbIil:'Iepe3J, H BbI'IHCJ1HeMorrpe,ll,eJIeHHbIil: HHTerpa.rr:

1
J = /[(3 + t) dt - 5(2 + t)(3 - 5t)(-1 + t) dt + (3 -			5t)2(-1 + t)dt] =
o		1	45t2 + 50t3 ) dt = 9. •
		= /(24 - 25t -
		o
4.2.20. BbI'IHCJ1HTbKPHBOJIHHeil:Hblil: HHTerpa.rr
		/ xydx - ydy
		L
B,ll,OJIb KPHBOil: L = CD, COe,ll,HHHIOIIIeil: TO'l~H C(4,0) H D(0,2),
eCJ1H:
1)	CD -	OTpe30K rrpHMoil:j
2)	CD -	rrapa6oJIa, cHMMeTpH'IHMOTHOCHTeJ1bHO OCH OXj
3)	CD -	rrapa60JIa, cHMMeTpH'IHMOTHOCHTeJIbHO OCH Oyj
4)	CD -	,ll,yra 9JIJIHnCa C n;eHTpOM B Ha'la.rreKOOp,ll,HHaT.

BW"'UCJI(Umb npocme~ut'IJ.M o6pa30M oaHHwe UHmeepaJl.W om nOJl.HWX oug)(pepeHqUaJ&oe:

	(2,3)			(3,4)
4.2.21.	/	xdy+ydx.	4.2.22.	/	x dx + y dy.
	(-1,2)			(0,1)
	(1,1)			(2,1)
4.2.23.	/	(x + y)(dx + dy).	4.2.24.	/	ydx ~xdy , y # o.
	(0,0)			(1,2)	Y

209