Сборник задач по высшей математике 2 том
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3.2.17. |
II Ja2 - |
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y2 dxdy, r,ll;e D - nOJIyKpyr X2 + y2 ~ a2, y ~ 0. |
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3.2.18. |
I dx |
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Jx2 + y2 dy. |
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3.2.19. |
II Ja2 - |
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y2 dxdy, r,ll;e D OrpaHHqeHa JIeMHHcKaToit |
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3.2.20. II Jl- :: - ~:dxdy, r,ll;e D- BHYTpeHHOCTb 9JIJIHnCa
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3.2.21. |
II f(x, y) dxdy, r,ll;e D OrpaHHqeHa OKPY:>KHOCT1.IMH x 2 + y2 = 4x, |
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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. |
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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, |
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X = 0. |
3.2.24. |
II ';'x-y-_-y"""2 dxdy, r,ll;e D - TpaneU;H1.I C BepIIIHHaMH A(I,I), |
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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 |
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(x> 0). |
3.2.26. |
II (X + 2y) dxdy, r,ll;e D OrpaHHQeHa KpHBbIMH y = x 2, Y = ..jX. |
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3.2.27. |
II J9 - x 2 - y2 dxdy, r,lJ;e D OrpaHH'IeHaKpHBbIMH |
y = x, y = |
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= V3x, x 2 + y2 = 9. |
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3.2.28. |
II (x2+y2) dxdy, r,lJ;e D OrpaHH'IeHaOKPy:>KHOCTbIO x 2+y2 = 2Rx. |
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3.2.29. |
II dxdy, r,lJ;e D OrpaHH'IeHaJIHHHeii(x 2 +y2)2 = 2ax3 . |
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3.2.30. |
II JR2 - x 2 - y2 dxdy, me D - Kpyr x 2 + y2 ~ Rx. |
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3.2.31. |
II y dxdy, me D - nOJIyKpyr (x - a)2 + y2 ~ a2, y ~ O. |
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3.2.32. |
II (x 2 + y2) dxdy, r,lJ;e D - |
Kpyr x 2 + (y + 2)2 ~ 4. |
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3.2.33. |
II arctg ~ dxdy, r,lJ;e D - |
'IeTBepTbKpyra x 2 + y2 |
~ 1, x ~ 0, |
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Y ~ O. |
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3.2.34. |
II dxdy, me D OrpaHH'IeHaJIeMHHCKaTOii (x2 + y2)2 = 2a2xy. |
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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.
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3.2.41. |
JJI(x, y) dxdy, 06JIaCTb D - o6rn;ruI 'IaCTbKpyroB x 2 + y2 ::;; ax, |
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x 2 + y2 ::;; bx. |
3.2.42. |
JJI(x, y) dxdy, 06JIaCTb D OrpaHH'IeHaIIpHMbIMH y = -x, y = x, |
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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.:
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3.2.43. |
Jdx |
JI(x, y) dy (a> 0, a > 0), eCJIH U = x, v = ~. |
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3.2.44. |
Jdx |
JI(x, y) dy, eCJIH U = x + y, V = X - y. |
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3.2.45.JJI(x, y) dxdy, me D - 06JIaCTb, OrpaHH'IeHHaHKPHBOii
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(X2 + ly2)2 = x 2y,
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eCJIH x = r cos <p, y = V3r sin <po |
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3.2.46. |
JJI(x, y) dxdy, r,!l;e D OrpaHH'IeHaIIapa60JIaMH y = ax2, y = by2 |
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eCJIH y = ux2, xy = v (0 < a < b, |
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H rHIIep60JIaMH xy = p, xy = q, |
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0< p < q). |
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3.2.47. |
TIpeo6pa30BaTb C IIOMOlII,bIO IIO,!l;CTaHOBOK x = ar cos <p, y = br sin <p |
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HHTerpaJI |
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JJ1(R2 - :~ - |
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r,!l;e D - JIe)Karn;ruI B IIepBoii 'IeTBepTH 'IaCTb 9JIJIHIITH'IeCKOrO |
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3.2.48. |
Jdx |
JI(x, y) dy. |
3.2.49. |
J |
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3.2.50. |
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JI( Jx 2 + y2) dy. |
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3.2.51.B ,!l;BOtiHOM HHTerpaJIe
IIf(x,y)dxdy,
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r,!l;e 06JIaCTb D OrpaHH'IeHaKpHBbIMH ..;x+vY = va, x = 0, y = 0, |
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C,!l;eJIaTb 3aMeHY nepeMeHHblx x = U cos4 V, Y = U sin4 v. HHTerpaJI |
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npHBeCTH K nOBTopHOMY. |
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3.2.52. |
BbI'IHCJIHTb |
II dxdy, |
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x2 |
y2 |
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r,!l;e D OrpaHH'IeHaKPHBOti a2 |
+ b2 = Ii |
+ k· |
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3.2.53. |
BbI'IHCJIHTb |
II dxdy, |
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Vf = 1, x = 0, y = o. |
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3.2.54. |
BbI'IHCJIHTb |
II r21sin (<p + ~f) - rl |
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drd<p, |
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r,!l;e D - npHMoyrOJIbHHK 0 ::;; r ::;; 1, |
0::;; |
<p ::;; 21f. |
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§ 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
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S = S(D) = Ildxdy. |
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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.
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3. TIYCTb V - TeJIO, OrpaHH'IeHHOecBepxy 1I0BepXHOCTbIO Z = I(x, y), CHH3YnoaepxHocTbIO Z = g(x, y), IIpH'IeMIIpoeKIIHeil: 06eHx 1I0BepxHocTeil: Ha llJIOCKOCTb
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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.
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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.
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IIpHH:!ITbI TaK:lKe 0603Ha'leHH:!I:f~(x, y) = p, f;(x, y) = q. B TaKOM CJIY'Iae,
S = //..)1 + p2 +q2 dxdy.
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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
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(<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;
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J o = Jx +Jy |
= //(x 2 +y2)p(x,y)dxdy. |
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B CJIY'IaeOP;HOPOP;HOit IIJIacTHHbI (p = 1) 9TH <P0PMYJIbI npHHHMaIOT 60JIee |
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npoCToit BHP;: |
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Jx = //y2 dXdY , J y = |
//x 2 dxdy, |
Jo = / /(x 2 + y2) dxdy. |
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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- |
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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 |
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Xc = ----:,-:--- |
Yc = ----:,-:--- |
!!dXdy' |
!!dXdy |
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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-
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6epeM TaK, KaK YKa3aHo Ha 'IepTe)Ke(pHC. 24).
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Pu.c. 24 |
CHa'IaJIaonpe,IJ;eJIHM KOOp,IJ;HHaThI TO'IKHA: |
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y2 = 2x |
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= 2x |
=> Xl = 0, YI = °H X2 = 2, Y2 = 2. |
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y =x, |
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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 |
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( X2 + y2)2 = 2xy . |
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a2 |
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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
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o<p |
= lac?s<p |
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bsm<p |
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KPHBaH B IIOJIHPHbIX KOOp)];HHaTaX HMeeT BH)]; |
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= |
2abr2 sin <pcos <p |
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T. e. (r2)2 = |
abr2 . 2 sin <p cos <p |
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...;ab . |
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= -e-y'sm2<p. BHYTpeHHocTb IIe- |
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~ ~y'sin2<p, IIpH 9TOM sin 2<p ~ 0, T. e. |
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o~ <p ~ ~. TaKHM 06Pa30M, |
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s = II dxdy = II abrdrd<p = |
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Vab = |
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11" |
11" |
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(21-oVSID2CP) |
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= ab I d<p |
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r dr = ab I d<p |
r2 0 c |
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11" |
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11" |
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"2 |
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= at I |
~~sin 2<pd<p = ~; |
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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
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HeIIpHeMJIeMbI BBH.n:y CJIO)l{HOCTH IIpe,n;eJIOB HHTerpHpOBaHH5I. llpoH3Be,n;eM 3aMeHY IIepeMeHHblx IIO <p0pMYJIaM
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{ 4X -7y + 8 = u |
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X = 53 (8u + 7v + 1) |
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ax 8 |
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53' T.e. |
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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- |
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S = II dxdy = II J dudv = II 53 dudv = 53 S (G) = 6:;. |
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3.3.15. BblQHCJIHTb IIJIOrn;a,n;b <PHrYPbI, OrpaHHQeHHoil: KpHBbIMH |
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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) |
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acos <p |
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o |
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11" |
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I (1 + 2 cos <p) d<p + a2 1(1 + 2 cos <p + cos2 <p) d<p = |
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= a' [1(1 + 2 cos,,) d<p +1(1+ 200',,) d<p +1cos' "d,,] |
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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
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Puc. 26 |
Puc. 27 |
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06'beMTeJIa BbI9HCJIHM no <p0pMYJIe |
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V = / / (4 - x) dxdy, |
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r,n;e 06JIaCTb D H306prur<eHa Ha pHC. 27. IIMeeM |
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V = /(4 |
2,rz |
2,rz |
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~x~) 14 = 128 |
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3.3.23. |
BbI9HCJIHTb 06'beMTeJIa, orpaHHgeHHoro nOBepXHOCTflMH Z = 0, |
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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