Конспект лекций по математическому анализу. Шерстнев А.Н. - 88 стр.

  x50. sWOJSTWA INTEGRALA, SWQZANNYE S NERAWENSTWAMI
  1. eSLI f; g INTEGRIRUEMY NA OTREZKE [a; b] I f (x)  g (x)(a  x  b),
   Zb           Zb
TO a f (x)dx  a g(x)dx.
   2.   eSLI f INTEGRIRUEMA NA [a; b], TO
          Zb          Zb
             f (x)dx  jf (x)jdx  K (b , a); GDE K = sup jf (x)j:
           a           a                              x2[a;b]
   3. pUSTX f (x)  0 (a  x  b); fZ INTEGRIRUEMA NA [a; b] I NEPRERYWNA
                                      b
W TO^KE c 2 [a; b]; f (c) > 0. tOGDA a f (x)dx > 0.
  p.1 SLEDUET IZ SRAWNENIQ SOOTWETSTWU@]IH INTEGRALXNYH SUMM rI-
MANA. dALEE IZ NERAWENSTW ,jf (x)j  f (x)  jf (x)j S U^ETOM P. 1 IMEEM
           Zb        Zb          Zb Zb           Zb
         , a jf j = a (,jf j)  a f  a jf j  a K = K (b , a);
I P. 2 DOKAZAN. pEREJDEM K P. 3. pUSTX DLQ OPREDELENNOSTI a < c < b.
w SILU 22.4 SU]ESTWUET OKRESTNOSTX U (c) = (d; e) (a  d < c < e  b)
TAKAQ, ^TO 0 <   f (x)(x 2 U (c)). tOGDA
                Zb Zd Ze Zb Ze
                    = + +   (e , d) > 0: >
                 a      a     d    e    d
     4. t E O R E M A [O SREDNEM ZNA^ENII]. pUSTX f; ' INTEGRIRUEMY NA
[a; b]; '(x)  0 (a  x  b). tOGDA
                        Zb                Zb
                         a
                           f (x)'(x)dx =  '(x)dx;
                                           a
GDE m    M (m = x2inf[a;b] f (x); M = sup f (x)). eSLI, KROME TOGO, f
                                         x2[a;b]
NEPRERYWNA, TO SU]ESTWUET  2 [a; b] TAKOE, ^TO
                   Zb                      Zb
                      f (x)'(x)dx = f () '(x)dx:
                    a                       a
 '(x)  0 WLE^ET m'(x)  f (x)'(x)  M'(x) (a  x  b); INTEGRIRUQ
\TI NERAWENSTWA, IMEEM
                Zb         Zb                Zb
              m '(x)dx  f (x)'(x)dx  M '(x)dx:
                 a          a                 a

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