А.Н.Шерстнев - Математический анализ
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SU]ESTWUET TTOGDA 8" > 0 |
9c < b 8x; y 2 (c; b) |
(j F (x) , F (y) j< "). |
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oSTALOSX ZAMETITX, ^TO |
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j F (x) , F (y) j=j Za |
,(Za +Zx ) j=j |
Zx |
f (t)dt j : |
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4. [fORMULA nX@TONA-lEJBNICA]. pUSTX F |
: [a; b) ! R NEPRERYWNA, |
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2 |
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+ , SU]ESTWUET F |
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lim F (x); F 0(x) (a < x < b) NE- |
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PRERYWNA, PRI^•EM OPREDELENO F0(a+). tOGDA Za F |
0(t) dt = F (b,) , F (a), |
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GDE INTEGRAL W LEWOJ ^ASTI RAWENSTWA, WOZMOVNO, IMEET OSOBENNOSTX |
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W PRAWOM KONCE. |
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eSLI INTEGRAL |
ZabF0(t)dt SOBSTWENNYJ, TO FORMULA DOKAZANA RANEE |
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(SM. 52.1). pUSTX IMEETSQ OSOBENNOSTX W TO^KE b. iZ FORMULY nX@TONA- |
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lEJBNICA DLQ INTEGRALOW rIMANA IMEEM |
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F (b |
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F (a) = |
lim [F (x) |
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F (a)] = |
lim |
xF 0(t)dt = |
bF0(t) dt: |
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x!b, |
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x!b, Za |
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Za |
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5. u P R A V N E N I E. nAPISATX FORMULU nX@TONA-lEJBNICA DLQ INTEGRALA S OSOBENNOSTX@ W LEWOM KONCE I DATX E•E WYWOD.
6. p R I M E R. fUNKCIQ F (x) = 2px NEPRERYWNA NA [0,1] I F 0(x) =
x,1=2 NEPRERYWNA NA (0; 1); F 0(1,) = 1, TAK ^TO Z 1x,1=2 dx = 2pxj10 = 2:
0
x128. iNTEGRALY OT NEOTRICATELXNYH FUNKCIJ
iZU^ENIE PRIZNAKOW SHODIMOSTI INTEGRALOW S OSOBENNOSTX@ NA^NEM• SO SLU^AQ INTEGRALOW OT NEOTRICATELXNYH FUNKCIJ.
1. w \TOM x WS@DU PREDPOLAGAETSQ, ^TO PROMEVUTOK [a; b), WOZMOVNO, NEOGRANI^EN SPRAWA, FUNKCII f(t); g(t) (a t < b) NEOTRICATELXNY, I INTEGRALY
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f(t) dt; Za |
g(t) dt |
IME@T OSOBENNOSTI W PRAWOM KONCE. w SLU^AE NEOTRICATELXNOJ FUNKCII f
BUDEM PISATX Z bf (t)dt < +1, ESLI \TOT INTEGRAL SHODITSQ. w UKAZANNYH SOGLA[ENIQH: a
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2. fUNKCIQ F (x) Za |
f(t) dt (a |
x < b) NE UBYWAET. pRI \TOM F (x) |
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OGRANI^ENA TTOGDA Za f(t)dt < +1. |
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3. pUSTX f (t) g(t) (a t < b). tOGDA |
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(A) Za g(t)dt < +1 ) Za f (t)dt < +1; PRI \TOM Za f (t)dt |
Za g(t) dt, |
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(B) ESLI Za f(t) dt RASHODITSQ, TO RASHODITSQ I Za g(t) dt. |
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4. pUSTX g(t) > 0 |
I lim f (t) |
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> 0. tOGDA INTEGRALY ( |
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) SHODQTSQ |
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t!b, g(t) |
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ILI RASHODQTSQ ODNOWREMENNO. |
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dOKAVEM P. 3(A) I ^ASTX P. 4 (OSTALXNYE UTWERVDENIQ { (!!)). |
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eSLI |
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g(t) dt < + |
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, TO SU]ESTWUET M = lim |
g(t) dt. iZ NERAWEN- |
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STWA Za |
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x!b, Zxa |
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f(t) dt |
Za |
g(t)dt (x < b) SLEDUET, ^TO |
Za |
f(t) dt M (x < b). |
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Za |
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x!b, Za |
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pO\TOMU |
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f(t) dt = |
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lim |
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f(t) dt |
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M < + . |
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w USLOWIQH P. 4 PUSTX " (0 < " < ) PROIZWOLXNO. tOGDA SU]ESTWUET |
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c < b, ^TO , " < |
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f(t) |
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< + " (c < t < b), TO ESTX ( , ")g(t) < |
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g(t) |
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Za |
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f (t) < ( + ")g(t) (c < t < b). pUSTX, NAPRIMER, |
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f(t) dt < + |
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. tOGDA |
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Zc |
f (t)dt < +1 I Zc |
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gb (t)dt , "Zc |
f (t)dt < |
+1. s U^ETOM• |
126.3 |
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OTS@DA SLEDUET, ^TO |
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Za g(t)dt < +1: |
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+1 |
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5. p R I M E R. |
Z1 |
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pOLOVIM W P. 3 f (x) = |
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x ; g(x) = |
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dx < + |
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e, (1 x < +1).g |
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6. u P R A V N E N I E. |
iSSLEDOWATX NA SHODIMOSTX SLEDU@]IE INTEG- |
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+1 |
arctg x |
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+1 x |
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RALY |
Z1 |
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dx; |
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x e, |
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dx ( ; > 0). |
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202
x129. sWQZX NESOBSTWENNYH INTEGRALOW S RQDAMI
1. ~ITATELX, NESOMNENNO, UVE ZAMETIL ANALOGI@ MEVDU INTEGRALAMI S OSOBENNOSTX@ I ^ISLOWYMI RQDAMI. oTMETIM RQD TO^NYH UTWERVDENIJ NA \TOT S^ET• . pUSTX INTEGRAL
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Za |
f (t)dt |
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IMEET OSOBENNOSTX W TO^KE b |
2 R[f+1g I POSLEDOWATELXNOSTX xn TAKOWA, |
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^TO a = x0 < x1 < x2 < : : : ; |
xn ! b. rASSMOTRIM RQD |
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j=1 Zxj,1 |
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2. eSLI INTEGRAL (1) SHODITSQ, TO SHODITSQ I RQD (2), PRI^•EM |
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(3) |
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f (t) dt = 1 |
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iMEEM |
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f (t)dt |
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lim |
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f (t)dt = lim[ |
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j=1 Zxj,1 |
j=1 Zxj,1 |
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Zxn,1 |
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P |
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P f(t) dt = |
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f (t)dt: |
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oBRATNOE UTWERVDENIE, WOOB]E GOWORQ, NEWERNO. oDNAKO:
3. eSLI f (x) |
0 (a x < b), TO IZ SHODIMOSTI RQDA (2) SLEDU@T |
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SHODIMOSTX INTEGRALA (1) I RAWENSTWO (3). |
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pUSTX s = j=1 Zxj,1 f (t)dt I x 2 (a; b) PROIZWOLXNO. tOGDA NAJDETSQ• |
n, |
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^TO x < xn, I SLEDOWATELXNOP |
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Za |
f(t) dt Za |
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f (t)dt; |
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TO ESTX Za |
f (t)dt jP=1 Zxj,1 f(t) dt. oSTA•ETSQ U^ESTX 128.2. |
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iNTEGRALXNYJ PRIZNAK SHODIMOSTI ^ISLOWOGO RQDA 59.1 MOVNO SFOR- MULIROWATX W TERMINAH INTEGRALA S OSOBENNOSTX@:
203
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4. eSLI FUNKCIQ f (x) |
0 (x 0) NE WOZRASTAET, TO INTEGRAL |
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Z |
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f(t) dt I RQD |
1 |
f(j) SHODQTSQ ILI RASHODQTSQ ODNOWREMENNO. |
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x130. aBSOL@TNO SHODQ]IESQ INTEGRALY |
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1. iNTEGRAL |
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(1) |
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Za |
f(t) dt; |
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IME@]IJ OSOBENNOSTX W PRAWOM KONCE, NAZYWAETSQ ABSOL@TNO SHODQ]IM-
Z bjf (t)j dt.
a
2. eSLI INTEGRAL SHODITSQ ABSOL@TNO, TO ON SHODITSQ.
b |
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pUSTX Za jf(t)j dt < +1. w SILU KRITERIQ kO[I 127.3 |
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y |
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8" > 0 9c < b 8x; y (c < x < y < b) (Zx |
jf (t)j dt < "): |
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y |
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y |
(t)j dt < ". sNOWA W SILU |
nO TOGDA DLQ UKAZANNYH x; y : |
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Zx |
f(t) dt |
Zx jf |
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KRITERIQ kO[I \TO OZNA^AET, ^TO INTEGRAL (1) SHODITSQ. >
kAK MY UWIDIM NIVE, IZ SHODIMOSTI INTEGRALA (1) EGO ABSOL@TNAQ SHODIMOSTX NE SLEDUET. pO\TOMU POLEZNO RASPOLAGATX PRIZNAKAMI SHODI- MOSTI BOLEE TONKIMI, ^EM PRIZNAKI DLQ INTEGRALOW OT ZNAKOPOSTOQNNYH FUNKCIJ. pRIWED•EM DWA POLEZNYH NA PRAKTIKE PRIZNAKA, KOTORYE W BOLEE OB]EJ FORME BUDUT DOKAZANY NIVE (SM. 135.4).
3. pUSTX b 2 R[ f+1g, INTEGRAL J = Z bf (t)g(t)dt IMEET EDINST-
WENNU@ OSOBENNOSTX W TO^KE b 2 R[ f+1g,a PRI^•EM f NEPRERYWNA, A g NEPRERYWNO DIFFERENCIRUEMA NA [a; b). pUSTX, KROME TOGO, WYPOLNENY USLOWIQ (PRIZNAK dIRIHLE)
1D) FUNKCIQ F (x) = Z xf(t) dt (a x < b) OGRANI^ENA,
a
2D) g(t) UBYWAET I lim g(t) = 0;
t!b,
LIBO WYPOLNENY USLOWIQ (PRIZNAK aBELQ)
1A) INTEGRAL Z bf(t) dt SHODITSQ,
a
204
2A) g OGRANI^ENA I MONOTONNA; | TOGDA INTEGRAL J SHODITSQ.
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+1 |
sin t |
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4. p R I M E R. iSSLEDUEM NA SHODIMOSTX Z0 |
t |
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dt. tAK KAK W LEWOM |
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KONCE OSOBENNOSTI NET, DOSTATO^NO ISSLEDOWATX NA SHODIMOSTX INTEGRAL |
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+1 |
sin t |
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Z1 |
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dt. pOLOVIM f (t) = sin t; g(t) = 1=t (t |
1), I MY NAHODIMSQ |
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W USLOWIQH PRIZNAKA dIRIHLE. iTAK, INTEGRAL SHODITSQ. oDNAKO ON NE |
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SHODITSQ ABSOL@TNO. |
f |
dOSTATO^NO POKAZATX (SM. 129.2), ^TO RASHODITSQ |
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u P R A V N E N I Q. sLEDU@]IE INTEGRALY ISSLEDOWATX NA SHODIMOSTX (W TOM ^ISLE ABSOL@TNU@):
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5. Z0 |
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6. |
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oTWET: PRI > 1 SHODIMOSTX ABSOL@TNAQ, |
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PRI 1 | NEABSOL@TNAQ.g
x131. nESOBSTWENNYE INTEGRALY (OB]IJ SLU^AJ)
dO SIH POR MY IMELI DELO S INTEGRALAMI, IME@]IMI EDINSTWENNU@ OSOBENNOSTX W ODNOM IZ KONCOW. pRIWED•EM TEPERX OB]EE OPREDELENIE.
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1. pUSTX a; b 2 R[ f 1g. fORMALXNYJ SIMWOL |
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NAZYWAETSQ NESOBSTWENNYM INTEGRALOM, |
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ESLI SU]ESTWUET RAZLOVENIE |
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c(ja = c0 < c1 < : : : < cn = b) |
TAKOE |
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^TO KAVDYJ IZ INTEGRALOW |
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Zcj,1 f (t)dt (1 j n) IMEET OSOBENNOSTX W ODNOM IZ KONCOW. pRI \TOM |
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INTEGRAL (1) NAZYWAETSQ SHODQ]IMSQ, ESLI SHODITSQ KAVDYJ IZ INTEGRA- |
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LOW Zcj,1 f(t) dt. w \TOM SLU^AE |
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205
eSLI HOTQ BY ODIN IZ INTEGRALOW Z cj f (t) dt RASHODITSQ, TO INTEGRAL (1)
NAZYWAETSQ RASHODQ]IMSQ.
cj,1
z A M E ^ A N I Q. 2. rAWENSTWO (2) KORREKTNO, TO ESTX EGO PRAWAQ ^ASTX NE ZAWISIT OT RAZLOVENIQ . fpOQSNIM \TO NA PRIMERE INTEGRALA (1) S DWUMQ OSOBENNOSTQMI W TO^KAH a I b. pUSTX a < c < c0 < b. tOGDA
c |
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+ Zc |
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+ |
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+ Zc0 |
! = |
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+ Zc |
! + Zc0 |
= Za |
+ Zc0 ; |
TAK KAK Z c0 f(t) dt | OBY^NYJ (SOBSTWENNYJ) INTEGRAL rIMANA.g
c
3. w SLU^AE INTEGRALA S OSOBENNOSTX@ WNUTRI PROMEVUTKA INTEGRIROWANIQ SLEDUET SDELATX ODNO PREDOSTEREVENIE. pUSTX INTEGRAL (1) IMEET EDINSTWENNU@ OSOBENNOSTX W TO^KE c (a < c < b). dLQ ISSLEDOWANIQ EGO NA SHODIMOSTX MY DOLVNY USTANOWITX SU]ESTWOWANIE PREDELOW
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|TO, ODNAKO, NE \KWIWALENTNO SU]ESTWOWANI@ PREDELA
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lim |
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# |
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KAK MOVET POKAZATXSQ NA PERWYJ WZGLQD. iZ SU]ESTWOWANIQ (3) SLEDUET SU]ESTWOWANIE (4) I ZNA^ENIE INTEGRALA (1) SOWPADAET S PREDELOM (4). oDNAKO IZ SU]ESTWOWANIQ (4) E]E• NE SLEDUET SU]ESTWOWANIE (3). sU]ESTWOWANIE PREDELOW (3) SLEDUET IZ SU]ESTWOWANIQ PREDELA
(5) |
lim |
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c," f(t) dt + |
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W SMYSLE PREDELA FUNKCIJ DWUH PEREMENNYH (SM. x66). tEM NE MENEE, ESLI (4) SU]ESTWUET, TO GOWORQT, ^TO INTEGRAL (1) SU]ESTWUET W SMYSLE GLAWNOGO ZNA^ENIQ (valeur principale):
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v.p. |
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lim |
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206
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aNALOGI^NO, ESLI INTEGRAL Z,1 f(t) dt IMEET OSOBENNOSTX LI[X NA KON- |
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, TO POD GLAWNYM ZNA^ENIEM PONIMAETSQ PREDEL (ESLI ON |
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SU]ESTWUET) v.p. |
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4. p R I M E R. iNTEGRAL Z,1 |
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RASHODITSQ, TAK KAK RASHODITSQ KAV- |
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DYJ IZ INTEGRALOW Z,1 |
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x132. kRATNYE NESOBSTWENNYE INTEGRALY
mY PRIWEDEM• NE SAMOE OB]EE OPREDELENIE KRATNOGO INTEGRALA S OSO- BENNOSTX@. oDNAKO EGO WPOLNE DOSTATO^NO DLQ BOLX[INSTWA PRILOVENIJ.
1.mNOVESTWO Rn NAZOWEM• LOKALXNO J-IZMERIMYM , ESLI J -IZMERIMO KAVDOE MNOVESTWO WIDA Br ( )\ (r > 0). o^EWIDNO, WSQKOE J -IZMERIMOE MNOVESTWO, BUDU^I OGRANI^ENNYM, LOKALXNO J-IZMERIMO.
2.pUSTX ( Rn )J -IZMERIMO I NEWYROVDENO (SM. 118.2), x0 2 , I f : R NE OGRANI^ENA NA , PRI^EM• DLQ L@BOGO " > 0 INTEGRAL!
Z |
f (x) dx OPREDELEN• KAK INTEGRAL rIMANA. fORMALXNYJ SIMWOL |
nB"(x0) |
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NAZYWAETSQ INTEGRALOM S OSOBENNOSTX@ W TO^KE x0. iNTEGRAL (1) NA-
ZYWAETSQ SHODQ]IMSQ, ESLI SU]ESTWUET PREDEL
(2) |
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f (x) dx |
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f (x)dx. eSLI PREDEL (2) NE SU]ESTWUET, |
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TO INTEGRAL (1) NAZYWAETSQ RASHODQ]IMSQ.
207
pUSTX TEPERX NEOGRANI^ENNOE MNOVESTWO (Rn) LOKALXNO J-IZMERI- MO, PRI^•EM MNOVESTWO Br ( )\ NEWYROVDENO, KOLX SKORO m(Br ( )\ ) > 0. pUSTX f : ! R INTEGRIRUEMA PO rIMANU PO L@BOMU MNOVESTWU WIDA Br( )\ . w \TOM SLU^AE (1) NAZYWAETSQ INTEGRALOM S OSOBENNOSTX@ W 1. iNTEGRAL (1) NAZYWAETSQ SHODQ]IMSQ, ESLI SU]ESTWUET PREDEL
r!+1 |
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pODOBNO ODNOMERNOMU SLU^A@ OPREDELQETSQ INTEGRAL S KONE^NYM ^ISLOM OSOBENNOSTEJ (NESOBSTWENNYJ INTEGRAL).
z A M E ^ A N I Q. 3. dANNOE OPREDELENIE SHODQ]EGOSQ INTEGRALA S OSO- BENNOSTX@ NE SWODITSQ K SOOTWETSTWU@]EMU OPREDELENI@ W ODNOMERNOM SLU^AE (126.1, 131.1). w SLU^AE OSOBENNOSTI WNUTRI PROMEVUTKA INTEG-
RIROWANIQ PRIWEDENNOE• |
ZDESX OPREDELENIE DAST NAM INTEGRAL W SMYSLE |
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GLAWNOGO ZNA^ENIQ. |
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4. eSLI INTEGRAL (1) |
SHODITSQ, TO |
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1. oTMETIM, ^TO INTEGRALY, STOQ]IE W LEWYH ^ASTQH PRIWEDENNYH• |
RA- |
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WENSTW, NESOBSTWENNYE. |
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oTMETIM NEKOTORYE SWOJSTWA WWEDENNOGO• PONQTIQ.
5.pUSTX INTEGRALY Z f (x) dx; Z g(x) dx IME@T EDINSTWENNU@ OSO-
,I SHODQTSQ. tOGDA SHODITSQ INTEGRALBENNOSTX W TO^KE x0 2 [f1g
Z [ f (x) + g(x)] dx, PRI^•EM
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[ f(x) + g(x)] dx = Z |
f (x) dx + Z g(x) dx ( ; 2 R): |
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6. eSLI f(x) 0 (x 2 |
) I SU]ESTWUET KONSTANTA C > 0 TAKAQ, |
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f(x) dx C PRI L@BOM " > 0, TO INTEGRAL (1) S EDINST- |
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WENNOJ OSOBENNOSTX@ W TO^KE x0 2 , SHODITSQ. (|TO SWOJSTWO LEGKO SFORMULIROWATX DLQ SLU^AQ x0 = 1.)
208
kRITERIJ kO[I |
SLU^AJ OSOBENNOSTI |
x0 |
2 |
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iNTEGRAL |
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SHODITSQ TTOGDA |
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8. gOWORQT, ^TO NESOBSTWENNYJ INTEGRAL (1) SHODITSQ ABSOL@TNO, ES- LI SHODITSQ INTEGRAL Z jf (x)j dx. oTMETIM, ^TO ESLI INTEGRAL SHODITSQ
ABSOL@TNO, TO ON SHODITSQ.
nAPRIMER, W SLU^AE EDINSTWENNOJ OSOBENNOSTI W TO^KE x0 2 , \TO |
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jf (x)j dx (r < s): |
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p R I M E R Y. 9. iNTEGRAL J = Z Z Z (x2 + y2 + z2), dxdydz( > 0), |
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GDE = f(x; y; z) : x2 + y2 + z2 1g, IMEET EDINSTWENNU@ OSOBENNOSTX W |
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TO^KE (0; 0; 0). pRI \TOM |
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iTAK, J SHODITSQ PRI < 3=2 I RASHODITSQ PRI 3=2: |
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10. wY^ISLIM |
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INTEGRALA: |
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y2 |
dxdy] |
1=2 |
= |
lim [ |
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11. u P R A V N E N I E. dLQ > 0 ISSLEDOWATX NA SHODIMOSTX INTEGRAL
Z Z Z (x2 + y2 + z2), dxdydz, GDE = f(x; y; z) : x2 + y2 + z2 |
1g. |
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209
integraly, zawisq}ie ot parametra
x133. nEPRERYWNOSTX SOBSTWENNYH INTEGRALOW PO
PARAMETRU
1. pRI SWEDENII KRATNYH INTEGRALOW K POWTORNYM MY WSTRE^ALISX S INTEGRALAMI WIDA
( ) F (x1; : : : ; xn) = Z f(x1; : : : ; xn; y1; : : : ; ym) dy1 : : : dym:
00
zDESX 00 Rm, A WEKTOR x = (x1; : : : ; xn ) NE ZAWISIT OT PEREMENNYH y1; : : : ; ym I IGRAET ROLX PARAMETRA. nASTOQ]IJ RAZDEL POSWQ]EN• IZU^E- NI@ FUNKCIJ, ZADANNYH INTEGRALAMI UKAZANNOGO WIDA (WOZMOVNO, NE- SOBSTWENNYMI). nAS BUDUT INTERESOWATX WOPROSY TAKOGO SORTA: BUDET LI
NEPRERYWNA FUNKCIQ F , ESLI NEPRERYWNA f? SU]ESTWUET LI @x@Fj , ESLI SU]ESTWUET @x@fj ? SPRAWEDLIWO LI RAWENSTWO
@x@Fj (x1; : : : ; xn ) = Z00 @x@fj (x1; : : : ; xn; y1; : : : ; ym) dy1 : : : d ym ?
I T. D. nA^NEM• IZU^ENIE SO SLU^AQ SOBSTWENNYH INTEGRALOW.
2. pUSTX MNOVESTWA 0 Rn; 00 Rm KOMPAKTNY, 00 J-IZMERIMO,= 0 00 ( Rn+m) I f : ! R NEPRERYWNA. tOGDA FUNKCIQ F, ZADAN- NAQ RAWENSTWOM ( ), TAKVE NEPRERYWNA.
uTWERVDENIE O^EWIDNO, ESLI m( 00) = 0. pUSTX m( 00) > 0: mNOVESTWO
0 00 KOMPAKTNO W Rn+m (SM. x107), TAK ^TO FUNKCIQ f RAWNOMERNO NE- PRERYWNA NA , TO ESTX (OBOZNA^AQ x = (x1; : : : ; xn ); y = (y1; : : : ; ym); z = (x; y) 2 Rn+m)
"
8" > 0 9 > 0 8z; z1 2 (kz , z1k < ) jf (z) , f (z1 )j < m( 00)): eSLI TEPERX kx , x1k < (x; x1 2 0), TO DLQ WEKTOROW z = (x; y); z1 =
210