.
I [, b] f(x). , , [, b] , , :
(1)
[, b] ,
ξi [i-1,_ i].
I -
I = F(b)-F(a), (2)
F(x) f(x).
, I . f(x); ,
F(x) f(x) , F(a) F(b) , f(x).
, (2) , - f(x). ( ) . : f()> 0 , [,b] f(x).
, .. (1). ≥ 1,
(3)
( , , [i-1, xi], f(ξi), . 1).
. 1. (.3)
(3) , ,, [, b] [xi-1, xi], ξi .
[, b] i
(4)
(3)
(5)
ξi [xi-1, xi]. .
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1) ξi = xi-1. (5)
(6)
2) (5) ξi = xi.
(7)
(6) (7) . (. 2), I .
..2. :
, ξi xi-1 xi. .
3) , ,
(8)
(8) (6) (7) , .
(6)(8) , , [xi-1, xi] , , h (4) [, b ], I ε. ( ) :
(9)
(9), , [, b), (8) h. (6), (.7) .
-
f(x) , -.
f(x )
, , (4) [, b] h, x=x0+qh
(10)
, Ln(x) f() ( = q= 0, =b q = ) , dx=hdq. ,
,
(11)
-,
(12)
.
, (11)-(12) . , .
.
n = 1. , [a, b] : x0 = a x1 = b.
. (12) n = 1 i = 0, 1 :
(13)
(13) (12) - :
(14)
h [a, b].
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f (x) ( 1- ). I(f, a, b) , ( ) f (x) (. . 3).
.3. .
- f (x),
- [x0, 1].
R, (14).
, (14), h. R (14).
, [a, b] N
h = (b - a) /N (15)
(14). , (15). yk (k = 0, 1, , N) , y0 yN. I(f, a, b) :
(16)
.
n=2. , [a, b] : 0=a, x1=(a+b)/2, x2=b. h=(b-a)/ 2
(11) n = 2 i = 0, 1, 2:
(7)
:
(18)
(b - a) = 2h.
(18) f(x) , (xi, yi), i = 0, 1, 2. (. . 4).
. 4. .
- f (x),
- L2(x) [x0, x2]. R (18).
, [x0, x1] , [x1, x2] - .
[a, b] N = 2M , M . : . M (18), .. . . i = 2 i = 2 M - 2, (18) , .
(17)
(19)
:
(20)
, So M , Se M-1 . h=(b - a)/(2M)=(b - a)/N.
(19) .
n = 3 :
(21)
h = (b - a) / 3.
[a, b] , (N = 3M), (20). :
(22)
h=(b-a)/(3M) = (b-a)/N, :
(23)
(22) . , .
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: 3 ǹ3.