Конспект лекций по математическому анализу. Шерстнев А.Н. - 209 стр.

   7.   [kRITERIJ kO[I (SLU^AJ OSOBENNOSTI x0 2                      , )].   iNTEGRAL (1)
SHODITSQ TTOGDA
                                                         Z                      
        8" > 0 9 > 0 8r; s r < s <  ) j                           f (x) dxj < " :
                                              [Bs(x0 )nBr (x0 )]\
   8. gOWORQT, ^TO NESOBSTWENNYJ
                      Z           INTEGRAL (1) SHODITSQ ABSOL@TNO, ES-
LI SHODITSQ INTEGRAL 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
SLEDUET IZ P. 7 I OCENKI
               Z                      Z
        j             f (x) dxj             jf (x)j dx (r < s):
          [Bs(x0 )nBr (x0 )]\        [Bs(x0 )nBr (x0)]\
                                          ZZZ
    p R I M E R Y. 9. iNTEGRAL J =                     (x2 + y2 + z2), dxdydz( > 0),
GDE = f(x; y; z) : x2 + y2 + z2  1g, IMEET EDINSTWENNU@ OSOBENNOSTX W
TO^KE (0; 0; 0). pRI \TOM
           ZZZ
     lim
    "!0+
                    (x2 + y2 + z2), dxdydz
            \B"()c
                              Z1              Z =2            Z 2        Z1
                      = "lim      2 ,
                                 r dr 2             cos 'd' dt = "lim    4 r2,2 dr.
                          !0+ "                ,=2             0    !0+    "
 iTAK, J SHODITSQZPRI < 3=2 I RASHODITSQ PRI  3=2:
                           +1 ,x2
    10. wY^ISLIM             e dx S ISPOLXZOWANIEM DWOJNOGO NESOBSTWENNOGO
INTEGRALA:               0
 Z +1 2                       Z NZ N 2 2                 1 = 2
                                                                    Z =2 Z N 2 1=2
        ,
       e dx = N lim
          x                  [          e , x  , y dxdy] = R!lim+1[ d' re,r ]
  0                    p!+1 0 0                                     0     0
                  = 2 :
     Z u P R A V N E N I E. dLQ > 0 ISSLEDOWATX NA SHODIMOSTX INTEGRAL
Z Z 11.
       (x2 + y2 + z2), dxdydz, GDE = f(x; y; z) : x2 + y2 + z2  1g.

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