Сборник задач по высшей математике 2 том
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3.1.51. |
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3.1.59. |
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3.1.61. |
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aeo1i1t'bte U1tme2pa.l/,'bt: |
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3.1.62. |
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X sin (x + y) dxdy, eCJIH D: 0 ~ x ~ 7r, 0 ~ Y ~ ~. |
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3.1.63. |
II x 2y COS(xy2) dxdy, eCJIH D: 0 ~ x ~ ~, 0 ~ y ~ 2. |
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3.1.64. |
II(X3+y3) dxdy, r.n:e D OrpaHHqeHaJIHHHHMH x-2y = 0, x-y = 0, |
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X = 4. |
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3.1.65. |
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D OrpaHHqeHaJIHHHHMH x = 0, y = 0, x = 1, |
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i.Y --""-::-3 dxdy, r.n:e |
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y=l. |
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3.1.66. |
II y2 sin x dxdy, r.n:e D OrpaHHqeHa JIl'(HHHMHx = 0, y = 0, x = 7r, |
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Y = 1 + cosx. |
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3.1.67. |
II y 2 sin2 xdxdy, r.n:e D OrpaHHqeHa JIHHHHMH x = -~, y = 0, |
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x=~,y=3cosx.
3.1.68.II (x + y3) dS.
1:(x:(2
O:(y:(2
3.1.69. II x: dxdy, r.n:e D OrpaHHqeHa JIHHHHMH x = 2, y = x, y = l.
D Y
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3.1.70. |
II xydxdy, r.n;e D - TpeyrOJIbHHK ABC C BepIIIHHaMHj A(O,O), |
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B(l,O), C(O, 1). |
3.1. 71. |
II y dxdy, r.n;e D OrpaHHGeHa JIHHHaMH y = 0, y = </X, y + x = 2. |
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Hatimu UHmeepa.n.bHoe cpeaHee 3Ha",eHue aaHHoti tjjYH'K:V,UU f(x, y) 6 Y'K:a3aHtwti o6.n.acmu D:
3.1.72. |
f(x, y) = eX+Yj D - |
KBa.u;paT °~ x ~ 1, °~ y ~ 1. |
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3.1.73. |
f(x,y) = sin2 X· sin2 Yj D - KBa.u;paT °~ x ~ 11', °~ Y ~ 11'. |
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3.1.74. |
f(x,y) |
= x 2 + |
2y2 + |
Xyj 06JIaCTb D OrpaHHGeHa JIHHHaMH x = 0, |
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y = 0, |
x + y = |
1. |
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3.1.75. |
f(x,y) = cos(x+y)j 06JIaCTbD OrpaHHGeHaJIHHHaMH x = 0, y = 11', |
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Y =x. |
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KOHTponbHble Bonpocbl 1ft 60nee CnO)l(Hble SaAaHlftR |
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3.1.76. |
IIpHBecTH npHMepbI <PYHKI.I.HH |
f(x, y), .n;JIa KOTOPOtt <popMYJIa H3 |
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TeopeMbI 0 cpe.n;HeM |
3HaGeHHH |
BepHa .n;JIa JIlo6ott TOGKH Mo H3 |
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06JIaCTH D. |
°He- |
3.1. 77. IIoGeMY B onpe.n;eJIeHHH .n;BottHoro HHTerpaJIa YCJIOBHe d -+ |
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JIb3a 3aMeHHTb YCJIOBHeM n -+ oo? |
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3.1.78.KaK MO)KHO C nOMOrn;blO .n;BottHoro HHTerpaJIa Bblpa3HTb o6beM Te-
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JIa, OrpaHHGeHHOrO cBepxy nOBepxHocTblO Z = f (x, y), a CHH3Y - |
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nOBepxHocTblO Z |
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g(x, y), 3a.u;aHHbIX Ha O.n;HOtt H TOtt )Ke o6JIa- |
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CTH D? [<DYHKI.I.HH f(x, y) H g(x, y) HenpepbIBHbI H f(x, y) ~ g(x, y) |
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T/(x, y) |
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3.1. 79. |
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3.1.80. |
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3.1.81. |
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3.1.82. |
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141
IIpeocma6um'b 6 6UOe n06mOpH'btX 060iJ.HOiJ. UHmeepa.ll
II !(X, y) dxdy,
D
ec.llu o6.11acm'b D OepaHU"teHa .IIUH'l.I.RMU:
3.1.83.y = _x2 + 3x, y = ~x.
3.1.84.y = 1 + sin x, y = -1, x = 0, x = 211'.
3.1.85.x 2 + y2 = 2a2, x 2 = ay (a> 0).
3.1.86.x 2 + y2 = ax, x 2 + y2 = 2ax, y = 0 (a> 0).
3.1.87. II (x + y) dxdy, D orpaHlPieHa JIHHlUlMH x = 0, y = x 2 + 2x - 3"
D
2y = 3x.
3.1.88.II xydxdy. 3.1.89. II (2x 2y-xy2)dxdy.
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(x_2)2+y2~1 |
O~x~l |
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3.1.90. |
II ~: dxdy, D OrpaHH'IeHaJIHHHj[MH y = lx, y = Vi, x = 1. |
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3.1.91. |
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3.1.92. |
II(x2 + y) dxdy, D OrpaHH'IeHaJIHHHj[MH x - 2y = 0, 2x - y = 0, |
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xy = 2.
OqeHUm'b UHmeepa.ll'bt:
3.1.93.II (x 2 + 4y2 + 10) dxdy.
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X2+y2~9 |
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II (x+y+xy)dxdy. |
3.1.94. |
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(x 2 + y2) dxdy. 3.1.95. |
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3Ixl+4lyl~12 |
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2~y~3 |
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3.1.96. |
(l-x 2 - y 2)dxdy. |
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(x-l)2+(y-l)2~1
IIo"teMY OaHH'bte o6oiJ.H'bte UHmeepa.ll'bt 3a6UCJlm om nop.Ro'K:a UHmeepup06a- H'l.I.R?
3.1.97. II (:: -2;a3)e-;2 dxdy.
O~x~l
O~y~l
142
3.1.98.
§2. 3AMEHA nEPEMEHHblX B ABO~HOM
VI HTErPAJ1 E
PacCMOTPHM ,n;BOil:HOil: HHTerpaJI
!!/(x,y)dXdy
D
B rrpl'IMoyrOJIbHbIXKoop,n;HHaTax (x, y). IIpe,n;rroJIO)KHM, 'ITOrrepeMeHHbIe X H Y l'IBJIl'I- IOTCl'I<PYHKIJ;Hl'IMH,n;BYX rrepeMeHHbIX U H V, T. e. X = X(U, V), y = y(U, V), H 3TH <PYHKIJ;IUI HerrpepbIBHbI BMeCTe CO CBOHMH '1aCTHbIMHrrpOH3BO,n;HbIMH rrepBOrO rrOpl'l,n;Karro
U H V B HeKoTopoil: 3aMKHYToil: 06JIacTH G rrJIOCKOCTH Ouv. IIpe,n;rroJIO)KHM TaK)Ke, QTO 3TH <PYHKIJ;HH B3aHMHO O,n;H03Ha'iHOH HerrpepbIBHO oTo6p~aIOT 06JIaCTb G Ha 06JIaCTb D.
Tor,n;a HMeeT MeCTO paBeHCTBO
!!/(x, y) dxdy = !!/[x(u, V), y(u, V)] 'IJ(u, v)1 dudv, r,n;e
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ax |
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au |
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J = J(u, v) = |
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BbIp~aeT 3JIeMeHT IIJIoru;a,n;H B 06JIacTH G, a K03<P<PHIJ;HeHT H3MeHeHHl'I3JIeMeHTa rrJIoru;a.n;H G rrpH npeo6pa30BaHHH
B 3JIeMeHT rrJIoru;a,n;H D.
Koop,n;HHaTbI ( u, v) Ha3bIBaIOTCl'I ",pu60.aU'lte'it'lt'bl.MU ",oopiJU'ltamaMU TO'lKH
(x,y), rrOCKOJIbKY ypaBHeHHl'Ix(u,v) = const H y(u,v) = const rrpe,n;CTaBJIl'IIOTHeKOTopbIe JIHHHH, Boo6ru;e rOBOpl'l,KpHBble, B 06JIacTH G.
HHTerpaJI
!!/[x(u,v),y(u,v)] 'IJ(u,v)ldudv
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l!a3bIBaeTCl'I,n;BOil:HbIM HHTerpaJIOM B KPHBOJIHHeil:HbIx Koop,n;HHaTax.
IIpocTeil:mHM H B~Heil:mHM '1aCTHbIM CJIY'IaeM KPHBOJIHHeil:HbIX Koop,n;HHaT aBJIl'IIOTCl'IrrOJIl'IpHble Koop,n;HHaTbI (T, 'P). OHH CBl'I3aHbI C rrpl'lMoyrOJIbHbIMH Ko-
143
op.n;HHaTaMH cpOpMyJIaMH X |
= rcos<p, |
y = rsin<p (r ~ 0, |
0 ~ <p < 271") . .HKo6HaH |
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npeo6pa30BaHHH B 3TOM CJIy'laepaBeH |
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J(r, <p) = |
or |
a<p |
= IC?S <p |
-rsin <pI |
=r |
, |
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ay |
ay |
sm<p |
rcos<p |
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or |
a<p |
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a dxdy = r drd<p - 3JIeMeHT nJIOw;a.n;H B nOJIHpHblX Koop.n;HHaTax.
I1PH 3TOM HMeeT MeCTO cpopMyJIa 3aMeHbI nepeMeHHblX B .n;BOil:HOM HHTerpaJIe npH nepexo.n;e K nOJIHpHblM Koop.n;HHaTaM
IIf(x, y) dxdy = IIfer cos <p, rsin <p) rdrd<p.
D G
K nOJIHpHblM Koop.n;HHaTaM oco6eHHO y.n;06HO nepexo.n;HTb B Tex CJIy'laHX,KO- r.n;a 06JIacTb HHTerpHpOBaHHH Kpyr HJIH '1aCTbKpyra. PaCCTaHOBKa npe.n;eJIOB H BbI'IHCJIeHHe.n;BotlHoro HHTerpaJIa B KPHBOJIHHetlHblx Koop.n;HHaTax BblnOJIHHeTCH
aHaJIOrH'IHOCJIy'l8.IOnpHMoyrOJIbHblX Koop.n;HHaT.
3.2.1. BhI'IHCJIHTh.n:BOttHOtt HHTerparr
II(2x + y) dxdy
D
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no 06JIaCTH D, OrpaHH'IeHHottnpHMhIMH y = 2x - |
3, y = 2x + 5, |
a |
y = -x + 7, y = -x-l. |
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06JIacTh D - naparrJIeJIOrpaMM ABCK (pHC. 19 a). |
XOTH no.n:hIHTe- |
rparrhHM <PYHKIJ.HH H 06JIaCTh HHTerpHpOBaHHH npOCThI, BhI'IHCJIeHHe.n:aHHOro HHTerparra B npHMoyroJIhHblX Koop.n:HHaTax .n:OCTaTO'lHOrpOM03.n:KO (y6e-
.n:HTeCh caMoCTOHTeJIhHo). 3aMeTHB, 'ITOypaBHeHHH npHMhIX MO)KHO 3anHcaTh B BH.n:e y - 2x = -3, y - 2x = 5, y + x = 7 H Y + X = -1, nepett.n:eM K HOBhIM Koop.n:HHaTaM, .n:JIH '1ero0603Ha'lHM
IIMeeM
{U=y-2X, |
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x:: 3 :v3' |
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v = y + x, |
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Y - |
3 + 3· |
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J= |
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2 |
-3' |
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T. e. IJI = l. |
au |
av |
3 |
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B HOBOtt CHCTeMe Koop.n:HHaT |
(u, v) |
06JIaCTh G OrpaHH'IeHa |
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npHMhIMH u = |
-3, u = 5, v = |
-1, v = 7, T. e. npe.n:CTaBJIHeT co6ott npHMQ- |
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yroJIhHHK (pHC. 196), a no.n:hlHTerparrhHM <PYHKIJ.HH paBHa |
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2x + y = 2 (-!! + Q) + (!! + 2v) = -!! + 1v |
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3· |
144
y
v
C1 "...,.,.,.,.,.,.,.,.,.,7;,.,.,.,.===K1
_all!III!III! I!I!II!I!!1!!11!5 u
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6
Puc. 19
OTMeTHM, 'ITOnepBaJI CHCTeMa q,0PMYJI, HanHCaHHaJI BbIwe, npeo6pa3yeT napaJIJIeJIOrpaMM ABCK B npHMoyrOJIbHHK A1 B1C1 K 1 , BTOPaJI CHCTeMa - Hao6opOT, npeo6pa3yeT npHMoyrOJIbHHK A1 B1C1 K 1 B napaJIJIeJIOrpaMM ABCK. IIpH 9TOM BH)J.HO, 'ITOHanpaBJIeHHe o6xo)J.a BepWHH o)J.HoA q,HrYPbI COOTBeTcTByeT npOTHBOnOJIO)KHOMY HanpaBJIeHHIO o6xo)J.a BepWHH
.ll:pyroA. lIMeHHo n09TOMY J < O. IIepexo)J.HM K BbI'IHCJIeHHHM:
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II (2x+y) dxdy = II (-~ + ~v)·ldudv = ~ |
I5du |
I(7 |
-u+4v) dv = |
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ABCK |
A1B1C1Kl |
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-1 |
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5 |
7 |
5 |
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= ~ Idu( -uv + 2V2)1_1 = ~ |
1[(-71.1. + 98) - |
(1.1. + 2)]du = |
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=1 !(-8u+96)du=1(-4u2+96u)1 |
= 704 . • |
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-3 |
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3.2.2. BbI'IHCJIHTb |
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I = II xydxdy, |
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r,n:e D - |
06JIacTb, OrpaHH'IeHHaJI KpHBbIMH y2 |
= |
4x, y2 |
= 9x, |
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a |
xy = 1, |
xy = 5. |
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06JIaCTb D H306paJKeHa Ha pHC. 20 a. 3aMeTHM, 'ITOpaCCTaBHTb npe)J.e- |
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JIbI HHTerpHpOBaHHH B HCXO)J.HOM HHTerpaJIe He npOCTO, o,n:HaKO no,n:xO)J.HIUaJI 3aMeHa nepeMeHHbIx n03BOJIHeT CBeCTH 9TOT HHTerpaJI K HHTerpaJIY no npHMoyrOJIbHHKy.
BBe,n:eM HOBbIe nepeMeHHbIe 1.1. |
H V npH nOMOIUH paBeHCTB y2 = ux, xy = V. |
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BbIpa3HM OTCIO,n:a nepeMeHHbIe x |
H y '1epe31.1. H v: x = |
3W |
= |
VUv· |
VU' y |
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145
y
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:::::.IU |
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4 |
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Puc. 20
oTKy.n;a, C Y'leTOMTOro, 'ITOx > 0 Ha 06JIacTH D, a 3Ha'lHT,u= ~2 > 0,
HMeeM IJ(u,v)1 = 3~'
TaKHM 06pa30M, HCXO.n;Hblft HHTerpaJI B nJIOCKOCTH Ouv HMeeT BH.n;
II \R:. VUV· 3~dudv = ill *dudv.
G G
rpaHHu;a 06JIacTH G onHCbIBaeTCH JIHHHHMH u = 4 (TaK KaK o.n;Ha H3 <popMyJI npeo6pa30BaHHH HMeeT BH.n; y2 = ux, TO JIHHHH y2 = 4x B nJIOCKOCTH
Oxy COOTBeTcTByeT JIHHHH u = 4 B nJIOCKOCTH Ouv), u = 9, v = 1, v = 5
(pHc.206).
IIo9ToMY 06JIacTb G HMeeT BH.n; 4 ~ u ~ 9, 1 ~ v ~ 5 (T. e. npe.n;CTa-
BJIHeT co6oft npHMoyroJIbHHK), a npe06pa30BaHHblft HHTerpaJI BbI'IHCJIHeTCH HaMHoro npome:
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95' |
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I = 1 If '!!. dudv = 1 I du I v dv = lIn u 1• v215 = 8In |
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B'bt6uPaJI noaxoaJlt,que 3a.MeH'bt nepe.MeHH'btX, 6'bt"tUC,/I,Umb a6011H'bte uHmeepa- ,/I,'bt, 3aaaHH'bte 6 npJl.Moyeo,/l,bH'btX 'lCOOpaUHamax:
3.2.3. |
II(y-x) dxdy, r.n;e D OrpaHH'IeHaJIHHHHMHy = -iX+5, y = x+l, |
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y = x - 3, y = -"3x +"3' |
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II dxdy, r.n;e D - |
napaJIJIeJIOrpaMM co CTopOHaMH Ha npHMbIX |
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146
3.2.5. |
II .jXy dxdy, r.n;e D OrpaHH'ieHaKpHBbIMH |
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= ax, y2 = bx, |
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xy = p, xy = q (0 < a < b, 0 < p < q). |
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3.2.6. |
II(x+y) dxdy, r.n;e D OrpaHH'ieHanpHMbIMH x+y = 4, x+y = 12 |
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H napa6oJIoit y2 = 2x. |
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3.2.7. |
BbI'iHCJIHTbHHTerpaJI |
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I = II Jr4a--:2:---x2~_-y~2dxdy, |
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D |
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Kpyr x 2 + y2 ~ 2ax. |
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o CTPOHM Kpyr x 2+y2 ~ 2ax pa,nHyca a C u:eHTpOM B TO'iKe(a, 0) (pHC. 21). |
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Ilo.n;bIHTerpaJIbHM |
¢YHKU:HH 'ieTHM no nepeMeliHoit |
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=f(x, y)), a 06JIacTb HHTerpHpoBaHHH CHMMeTpH'iHaOTHOCHTeJIbHO OCH Ox.
Il09ToMY MO}l{HO BbI'iHCJIHTbHHTerpaJI TOJIbKO no BepxHeMY nOJIyKpyry H
pe3YJIbTaT y.n;BOHTb:
I = 2 IIJ 4a2 - x 2 - y2 dxdy.
D/2
x p
Puc. 21
IIepexo.n;HM K nOJIHpHbIM Koop.n;HHaTaM x = r cos cp, y = r sin cpo ,nJIH
y.n;06cTBa paCCTaHOBKH npe.n;eJIOB B nOJIHpHbIX Koop.n;HHaTax COBMeCTHM noJIHpHyIO cHcTeMY C npHMoyroJIbHoit TaK, KaK 9TO nOKa3aHO Ha pHC. 21. To- r.n;a nOJIyKpyr D /2 B nOJIHpHbIX Koop.n;HHaTax 3a,naeTCH CHcTeMoit HepaBeHcTB
o~ cp ~ ~, r |
~ 2a cos cp, no.n;bIHTerpaJIbHM ¢YHKU:HH npHMeT BH.n; v'4a2 - r2, |
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a dxdy = r drdcp. TaKHM 06pa30M, |
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2a cos <p |
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J4a2 - r2. (-~) d(4a2 - r2) = |
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1[(4a2 - 4a2 cos2 <p) ~ - (4a2 ) ~] d<p = |
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1[(4a2sin2<p)~ -8a3 ]d<p= i |
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3.2.8. |
11 |
Jx2 + y2 dxdy. |
3.2.9. |
11 |
Jl - x 2 - y2 dxdy. |
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z2+y2~a2 |
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z2+y2~1 |
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sin Jx 2 + y2 dxdy. |
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BbI'U[CJIHTbnOBTopHblit HHTerpa.rr I |
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Q CHaT.Ja.rra npeo6pa3yeM nOBTopHbIit HHTerpa.rr B .n;BOitHoit:
1= ii eZ 2+ Y2 dxdy, |
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{O ~ X |
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o~ y ~ ..ja2 - x 2 • |
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a ............................................. |
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.:::::!:!:::!.!:!..:!.~!:!:::::.:.:!:::! |
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cp |
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Puc. 22 |
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06JIacTb HHTerpHpoBaHH8 |
npe.n;cTaBJI8eT |
co6oit |
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(pHC. 22 a), n09TOMY y.n;06HO nepeitTH K nOJI8pHbIM Koop.n;HHaTaM (r, <p). ITo- JI8pHyIO cHcTeMY Koop.n;HHaT H306pa3HM TaK)Ke B BH.n;e np8MoyrOJIbHoit
(pHC. 22 6). Tor.n;a 06JIaCTb G B CHCTeMe Koop.n;HHaT Or<p onpe.n;eJI8eTC8 CH- |
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cTeMoit HepaBeHCTB |
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np8MoyroJIbHHK. Y'lTeMTaK)Ke, GTO no.n;bIHTerpa.rrbHa8 <PYHKIJ.H8 |
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HMeeT BH.n; er |
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CJIe.n;OBaTeJIbHO, |
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cos cp+sm |
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o=~(ea |
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.'2er |
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Ja2 _ y 2 |
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3.2.12. |
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Ja2 - |
y2 - X 2dx. |
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3.2.13. |
II J X 2 + y2 - 9 dxdy, D - KOJIbI.J,O, OrpaHlf'ieHHOeOKPJ)KHOCTjI- |
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MH X 2 + y2 = 9 H x 2 + y2 = 25. |
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3.2.14. |
Idy |
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Ja2 - x2 _y2dx. |
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3.2.15. |
II (x 2 + y2) dxdy, |
r.n:e 06JIacTb D OrpaHH'ieHa OKPY)KHOCTjlMH |
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x2 + y2 = ax, x2 + y2 = 2ax H OCblO Ox (y ~ 0).
3.2.16.BbI'iHCJIHTb
I = II xJx2 +y2dxdy,
D
r.n:e D - 06JIacTb, OrpaHH'ieHHMJIeMHHcKaToit
(x2 + y2)2 = a2(x 2 _ y2), X ~ o.
Puc. 23
o 3aMeHjIjI |
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Ha r cos cp, |
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r sin cp, |
IIOJIY'iHM Ha ypaBHeHHe JIeMHHCKa- |
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Thl |
(pHC. |
23) |
B |
IIOJIjlPHbIX |
Koop.n:HHaTax r |
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aJcos 2cp |
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(cos 2cp ~ 0 IIpH |
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- ~ |
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~ ~). IIo.n:bIHTerpaJIbHM <PYHKI.J,HjI paBHa r2 cos cpo B |
CHJIY CHM- |
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MeTpHH JIeMHHCKaTbI OTHOCHTeJIbHO OCH OX H 'ieTHOCTHIIo.n:bIHTerpaJIbHoit |
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<PYHKI.J,HH OTHOCHTeJIbHO IIepeMeHHoit y MO)KHO 3aIIHcaTb: |
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avcos2~ |
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1= 2 II r2 cos cp·r drdcp = 2 I |
cos cp dcp |
I r3 dr = ~a4 I |
cos2 2cp·cos cp dcp = |
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000 |
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2' |
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= ~ |
1(1- 2sin2 cp)2 d(sincp) |
= ~ 1(1- 4sin2 cp + 4sin4 cp) d(sincp) = |
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a4 |
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4. 5 |
cp |
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I~ |
2..;2 |
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= '2 |
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smcp- |
3sm |
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cp+ gsm |
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149