1. . , f (x) ³ 0 [ a,b ], , y = f (x), x = a, x = b.
(5.23)
[ a,b ] , . 5.6, , .
.
, [ , ] Ì [ a, b ] f (x) < 0,
1. , [0,2p].
. sin(x) ³ 0 [0, p] sin(x) £ 0 [p,2p], S
S =
= - (cosp - cos0) + (cos2p -cosp) = -(-1 1) +(1 + 1) = 4.
. 5.6.
. 5.7. .
, . . , .5.7
2. , y 1=½ x - 2½ y 2 = (. 5.8).
. 5.8. .
. .
y 1() = y 2().
; .
, .
. f (x) [ a,b ] AB. , .5.9 n M 0, M 1,..., M n. . , AB . D l i, i Mi + 1, i (xi, f (xi)), i +1 (xi + 1, f (xi +1))
.
i Mi +1 D l i
. (5.24)
(5.24) , L
. 5.9. .
.
, . , , r , . L
(5.25)
: = j(t), y = y(t) (a£ t £b),
|
|
(5.26)
1. y 2 = x 3 , x = 0 x = 1 (y ³ 0).
. . (5.25),
.
2. x = a cos3 t, y = a sin3 t, t 0 p/2.
. t
x ¢(t) = -3 a cos2 t ּsin t, y ¢(t) = 3 a sin2 t ּcos t
(5.26),
. , (.5.10), = = b. V. [ a, b ] n ,
x 0 = a < x 1 < x 2 <... < x i -1< x i <... < x n = b.
. 5.10. .
, . n ( ) Δ x i = x i - x i-1 (i = 1, 2,, n). Δ V i. [ x i-1, x i] . S (x *i) .
(5.27)
(5.27) i,
(5.28)
n. Δ x i, .. r . V,
(5.29)
(. 5.11), . V
(5.30)
. 5.11. .
. , y = sin(x) .
.
.