. [ a, b ] f (x). XOY, .5.1, , x = a y = b , S. [ a, b ] n ,
x 0 = a < x 1 < x 2 <... < x i -1< x i <... < x n = b.
, . n ( ) Δ x i = x i - x i-1 (i = 1, 2,, n). Δ S i.
[ x i-1, x i] , . i - , .
. 5.1. .
.
S . (5.1)
n. Δ x i. r, .. r = maxΔ x i 0 n ¥.
= S (5.2)
, (5.2) .
. [ a,b ] f (x). [ a, b ] n ,
x 0 = a < x 1 < x 2 <... < x i -1 < x i<... < x n = b.
[ x i-1, x i] (i = 1, 2,, n) , Δ x i = x i - x i-1, . i,
n. Δ x i, .. . [ a,b ] , f (x) n ¥ r 0, f (x) [ a,b ], f (x) - [ a,b ].
" ", S - Summa.
(5.3)
: " a b f (x)". b . (.. [ a, b ] ).
|
|
- ! f (x) , , .
, .
1. (5.4)
2. (5.5)
3. . (5.6)
4. . (5.7)
. f (x) [ a, c ] [ c, b ], a < c < b, [ a, b ],
. (5.8)
: , , (. 5.2). . f (x) - , .. f (- x) = - f (x),
(5.9)
f (x) - , .. f (- x) = f (x),
(5.10)
. 5.2. .
.
1. ,
(5.11)
,
.
2. .
(5.12)
. [ a, b ] f (x) g (x) (. 5.3).
.
. 5.3. . .
.
, , M m . ,
. f (x) [ a,b ], ,
. (5.13)
(.5.4): , , f () , b. f (c) [a,b]
. 5.4.