Конспект лекций по математическому анализу. Шерстнев А.Н. - 94 стр.

WELI^INY D (f ) = inf  

                          S (
); D(f ) = sup S(
). oNI NAZYWA@TSQ SOOT-
                                            

WETSTWENNO WERHNIM I NIVNIM INTEGRALAMI dARBU FUNKCII f . pRI \TOM
(P. 3) D (f )  D (f ).
    x56. kRITERIJ dARBU INTEGRIRUEMOSTI PO rIMANU
    1. oGRANI^ENNAQ FUNKCIQ f (x) (a  x  b) INTEGRIRUEMA PO rIMANU
        Zb
(I = f (x) dx) TTOGDA D (f ) = D(f )(= ).
         a
 nEOBHODIMOSTX. pUSTX f INTEGRIRUEMA. w SILU 46.5
 8" > 0 9
(a = x0 < x1 < : : : < xs = b) 8j 2 [xs j,1; xj ] (j = 1; s)
                                              (j P f (j )(xj , xj,1) , j < ").
                                                 j =1
 sLEDOWATELXNO, DLQ L@BYH j 2 [xj,1; xj ] (j = 1; s)
                              Xs
                        , " < f (j )(xj , xj,1) < + ":
                               j =1
wZQW sup (SOOTWETSTWENNO inf ) PO j W KAVDOM IZ OTREZKOW, POLU^IM
  , "  S(
)  S (
)  + ". oTS@DA S (
) , S(
)  2", I ZNA^IT,
D (f ) , D(f )  2". iZ PROIZWOLXNOSTI " : D (f )  D(f ) I OSTAETSQ
U^ESTX NERAWENSTWO W 55.4.
    dOSTATO^NOSTX. pUSTX D(f ) = D(f ) = . dLQ PROIZWOLXNOGO
" > 0 (W SILU OPREDELENIQ 55.4 I SWOJSTWA 55.3) NAJDETSQ RAZLOVENIE
~ a = x0 < x1 < : : : < xs = b) TAKOE, ^TO

(
                                 Xs
              S (
) , S(
) = (Mj , mj )(xj , xj,1) < " :
                 e        e
                                                                      2
                                      j =1
sLEDUET LI[X UBEDITXSQ, ^TO DLQ L@BOGO RAZLOVENIQ 
(a = y0 < y1 <
: : : < yN = b) DOSTATO^NO MALOGO DIAMETRA, MY BUDEM IMETX jS
 , j < ",
GDE S
 | INTEGRALXNAQ SUMMA rIMANA FUNKCII f . pUSTX M = sup jf (x)j
                                                              x2[a;b]
I d(
) < 4Ms " . mY IMEEM

          S
 , = P f (i )(yi , yi,1) ,
                       N
                      i=1
                   = P0f ( )(y , y ) + P00f ( )(y , y ) , ;
                           i      i          i,1        i   i   i,1
                      i                             i

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