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= lim

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#  $ 1+

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 $H!( (' )$(  0(  [x , x ] -, + ( 1+"%')> #$ ξ 4  ξ ∈ [x , x ] " "( (( σ = f (ξ )∆x 4 0! ∆x = x −x 5 ; (( % (



4 ' * !'& !0 +%," "& " !0 -,+ # $ ξ 5 ,%#"( # + % λ6 λ = max ∆x " % ( J #"' λ K   
5 k−1

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k−1

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k=1

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k=1

'& 0 H +%," "&  $#  ξ -,"+ ( "++9"')- #$" " ) "( " 0+')> (( σ 6 k

2

σ2 =

n


f (ξk )∆xk =

n


k=1

0 · ∆xk = 0.

k=1

#"4 σ %""  -,+ # $ ξ 4 0! σ  "(  1+ ! '4  %" &< 0  -,+ # $ ξ @!H !'& !0 +%," "&B "4 ' ! ')4 /$9"&  " 0+"+ (5 k

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 ∀ x , x ∈ X : |x − x | < δ.

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0

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ε δ |f (x ) − f (x )| = 2ξ|(x − x )| > 2 · = ε. δ 2

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7+ !1'H"(4 # y = f (x)  &'& & +( +  1+ +-*4 5 5 ∃ ε > 04 # !'& ∀ δ ∃ x , x ∈ [a, b] : |x − x | < δ 4  |f (x ) − f (x )  ε 5 -, + ( δ = 15 *!& #$" x , x ∈ [a, b]4 |x − x | < 1L |f (x ) − f (x )|  ε 5 -, + ( δ = 1/25 *!& %  +      
 ,

0



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x2 , x2 ∈ [a, b] 1 |x2 − x2 | < ; 2 1 δ= , 3

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1 , n

∃ xn , xn ∈ [a, b],

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1 ; n

@B

|f (xn ) − f (xn )|  ε0 ;

.........................

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 nk

{xnk } → c ∈ [a, b].

(+"( 1!1' ! ')) {x }5 -,"+ ( x   (" H n 4  0!  "'  +  |x − x | < 1/n 1' ! ')) {x } $ H + ("& $ c4 {x } → c5 >!4  "'  1+ +-" /$9"" f (x)   0(  [a, b] "( (  nk

 nk

 nk

 nk

k

 nk

k

 nk

{f (xnk )} → f (c),

{f (xnk )} → f (c)

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0


-   -( ,


1')%&  + ( +4 1 + *! ( $ +(+ "> -: $'  " 0+"+ (-: /$9"*5 &  %/ - #
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!!"( ∀ ε > 05
1')%&  ,:!"( " !# '" " 0+"+ (" /$9""4 1$H (4 # !'& J0 ε *! &  $ +%," " [a, b]  #"4 # ,!  -1'&)& S − s < ε5 #' 1')% (&  + (* + " 1 %!( ε 1!, + ( $ δ4 % "&<  ε " ')$  ε @δ(ε)B4 # |f (x ) − f (x )| < ε4 $$ ')$ |x − x | < δ @!'& ∀ x , x ∈ [a, b]B5 7+"% ! (  1 +) +%," " [a, b]  # " $4 #,- =0 +%," "& λ @λ = max ∆x B ,-' ( )= δ " "( +%) %  











k

1
k
n

S−s=



n


Mk ∆xk −

k=1

n


mk ∆xk =

k=1

n


(Mk − mk )∆xk .

k=1


%  1+ +-" /$9""   0(   [a, b] ' !   1+ +-)  x x '>,(  0(  [x , x ]4  1(  "' +*  + (-  * +=+  J(  0(   !"0  ": #-: 0+"94 5 5 M = f (ξ ), m = f (ξ )4 ξ , ξ ∈ [x , x ]5 r k−1

ξk r

ξk r

r k

k−1

 k

k

S−s =

n


(Mk − mk )∆xk =

k=1

< ε

@∆x

n


 k

k

n


 k

 k

k

k−1

k

[f (ξk ) − f (ξk )]∆xk <

k=1

∆xk = ε(b − a) = ε1

k=1

4 %#" |ξ − ξ | < δB5  + ( !$%5 #    + (  & 1+ !'"*4 '" /$9"& y = f (x)  1+  +-   : #$: [a, b]4 $+( ": $ #0 #"'4 0!   +1" +% +- 1 +0 +!5 k

<δ

 k

 k

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k=1

k

n


<

6

y = f (x)

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r x0 = a

x1

x2

r

x

-

b = xn

n


ε (Mk − mk )∆xk < (Mk − mk ) = f (a) − f (b) k=1

ε [f (b) − f (a)] = ε, f (b) − f (a)

=

$ $$ ∆x

y

ε f (b) − f (a)

"

(Mk − mk ) = M1 − m1 + M2 − m2 + . . . + Mn − mn =

k=1

= f (x1 ) − f (a) + f (x2 ) − f (x1 ) + . . . + f (b) − f (xn−1 ) = f (b) − f (a).

' ! ')4 0'  ,:!"(( " !#( '"> " 0+" + (" /$9"" @!#"B4 (- 1'#"(  +H! "  + (-5 5◦ 



a

5

f (x) dx = 0 a



 ,   
 


b

a f (x) dx = −

f (x) dx

5

 '" /$9"" f (x) " g(x) " 0+"+ (-  [a, b]4  /$9"" f (x)±g(x)4 a

c · f (x)

b

$H " 0+"+ (-  [a, b]4 1+"# (6 b

b [f (x) ± g(x)] dx =

a

b f (x) dx ±

a

g(x) dx, a

@B




b

b cf (x) dx = c

a

@DB

f (x) dx. a

$H ( 1 +  +H! " 5 #' 1$H ( "  0+"+ () /$9"" f (x)±g(x)5 '& J0 "( " 0+')> (( !'& J* /$9""6 %  

σ=

n


[f (ξk ) ± g(ξk )] · ∆xk =

n


k=1

f (ξk )∆xk +

k=1

n


g(ξk )∆xk .

k=1

$ $$ /$9"" f (x) " g(x) " 0+"+ (- @1 '">B4  , ( (-4 &<"  1+* #"4 "( > 1+ ! '- " "  %"&  1, +%," "&  0(  [a, b] "  -,+ # $ ξ 5 7J( ((4 &<&  ' * #"4 $ H "(  1+ ! '4 # " !$%-  " 0+"+ () /$9"" f (x) ± g(x)5 +( 04 $ $$ 1+ ! '- (( 1+* #"   +-    f (x) dx " g(x) dx4  1+ ! ' ((-4 &< * k

b

a

 ' * #" " +-* b

b

a

b

[f (x) ± g(x)] dx

4 ,!   !+0* +- + 

a

b

5 +  +H! " !$%- & '0"#5  '" /$9"& y = f (x) " 0+"+ (  [a, b]4 " [c, d] K #) [a, b]4 [c, d] ⊂ [a, b]4  /$9"& y = f (x) " 0+"+ (  [c, d]5 @ % !$% ') 5B  '" /$9"& y = f (x) " 0+"+ (  [a, c] " [c, b]4   "  0+"+ ( $H  [a, b] " 1+ !'" +  f (x) dx ±

a

g(x) dx

a

b

c f (x) dx =

a

b f (x) dx +

a

f (x) dx. c

@GB



.B 7+ !1'H"( #'4 # c ∈ (a, b)5 7$H ( # ' " 0+"+ () f (x)  [a, b]5 '& J0 %!!"( 1+"%') ε > 0 " 1$H (4 # *! & $ +%," " [a, b]4 !'& $+0 ,!  -1' &)& S − s < ε5 $ $$ f (x) " 0+"+ (  [a, c]4  !'& -,+0 ε *! & +%," " [a, c]  #"4 !'& $+0 S − s < ε/25 '0"#  [c, b]6 S − s < ε/25 ,M !"&& +%," "&  0(  [a, c] " [c, b]4 (- 1'#"( $ +%," " [a, b]  #"4 !'& $+0 ,!  "( ) (   +  %  









S − s = S  − s + S  − s <

ε ε + = ε. 2 2

7 + *! ( $ !$% ') +  @GB5 7+"% ! ( +%," " [a, b]  #" 1+"%')4  $4 #,- #$ x = c ,-' #$* ! ' "&5 0! # "! "(  (  +  σ=

0!

n


f (ξk )∆xk =

k=1

m


f (ξk )∆xk +

k=1

n


f (ξk )∆xk = σ1 + σ2 ,

k=m

K ((4 ' & !'& [a, c]L σ K ((4 ' & !'& [c, b]5 7+" λ = max ∆x → 0 σ " σ "( > $ #- 1+ ! '-4 +-       f (x) dx " f (x) dx5 ' ! ')4 σ H "(  1+ ! '4 σ1

2

k

c

b

a

b

1

2

c

+-* f (x) dx " "(  (  +  @GB5 B (+"( !+0 +1'H " # $ a < b < c5 $ $$ [a, b] ⊂ [a, c]4  " 0+"+ () f (x)  [a, b] ' !  "% " 0+"+ (" f (x)  [a, c] " * D5 +( 04 "% 1$ . * G ' !  a

c

b f (x) dx =

a

c f (x) dx +

a

f (x) dx. b



>!4 "1')%& * 4 1'# ( b

c f (x) dx =

c f (x) dx −

f (x) dx,

a

a

b

b

c

c

f (x) dx = a

f (x) dx + a

I " + ,') !$%)5

f (x) dx. b

        

 7) /$9"& y = f (x) " 0+"+ (  [a, b] " 1) f (x)  0

!'& ∀ x ∈ [a, b]5 0!

b

5

f (x) dx  0 a

"( 1+"%')> " 0+')> ((6 σ= f (ξ )∆x 5  "' '"&  + (- $H! '0 ( J* (( +"9 ')4 ' ! ')4  +"9 ')* ,!  " & (( σ4  ' ! ')4 " 1+ ! '4 +-* 1+ ! ' ( " 0+'5 %   n 

k=1

k

k


   $ f (x)

g(x)
   
 [a, b] b b  f (x)  g(x)  f (x) dx  g(x) dx a


 "

a

,%#"( # + % ϕ(x) = f (x)−g(x)5 $9"& ϕ(x)4 #  "!4 " 0+"+ (  [a, b] $$ +%) " 0+"+ (-: /$9"*5 + ( 04 1 '">  + (- f (x)  04 ' ! ')  "' 1+ !-!< 0 * ? %  

b

b ϕ(x) dx  0,

a

a

b

b f (x) dx −

a

[f (x) − g(x)] dx  0, b g(x) dx  0,

a

b f (x) dx 

a

g(x) dx. a



b a

 #"4 '" "(  (   +  f (x)  m4 ∀ x ∈ [a, b]4  f (x) dx  m(b − a)5  (( ! ' 4  "' ' !"& b

b f (x) dx 

a

m dx = m(b − a). a

#     ,  1 '" /$9"& y = f (x) 

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