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Oliy matmatika 2 kurs
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- 4-masala.
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3.19. 1) . 0
0 , 1 : , ) 2 ( 2 3 y x x y D dxdy y x D
2) . 0 , 0 , 0 , 9 : , 2 2 2 2 2 2 z y x z y x V dxdydz z y x V
. 2
: , 2 2 x y x y D dxdy xy D
2) . 0 , , 1 , 0 , 4 : , ) 2 1 ( z xy z x y x y V dxdydz z V
. 0
1 : , ) 2 ( 2 y x y D dxdy y x x D
2) . 2 1 , 3 0 , 1 0 : , ) 2 ( 2 2
z y x V dxdydz z y x V
. 4
, 2 2 2 2 y x D y x dxdy D
2) . 1 0 , 3 1 , 2 1 : , 3
z y x V yzdxdydz x V
. 9
, 2 2 2 2 2 2 y x D dxdy y x e D y x
2) . 0 , , 1 , 0 , : , ) 3 2 ( 3 2 2 z y x z x x x y V dxdydz x y V
. 3
3 : , ) 1 ( 2 2 y x y D dxdy y x D
2) . 0 , 0 , 0 , 1 : , ) ( z y x z y x V dxdydz z y x V
. 2
1 , : , 2 2 y xy x y D dxdy x y D
2) . 2 1 , 2 0 , 3 1 : , 3 2 2 z y x V dxdydz z y x V
4-masala. Egri chiziqli integrallarni hisoblang: 4.1. 1)
2 : 2 parabolaning y x 2 2 parabola kesgan yoyi; 2) L ydy x dx xy 2 ) 1 ( , : L
) 0 ; 1 ( A va
) 2 ; 0 (
nuqtalarni tutashtiruvchi
to‘g‘ri chiziq kesmasi. 4.2. 1)
dl x 2 , 2 2 2 : R y x L aylananing yuqori yoyi; 2)
L xdy dx y xy 2 ) ( ,
: L
2 x y parabolaning ) 0
0 (
nuqtadan ) 1 ; 1 ( B nuqtagacha bo‘lgan yoyi. 22
4.3. 1) L dl y x ) ( 2 2 , x y x L 4 : 2 2 aylana; 2)
dy y x y dx x y x ) 2 ( ) ( 2 2 , t y t x L sin
2 , cos 3 : ellipsning musbat yo‘nalishda aylanib o‘tishdagi yoyi.
L dl y x ) ( , : L uchlari ) 0 ; 0 ( ), 1 ; 0 ( ), 0 ; 1 ( O B A nuqtalarda bo‘lgan uchburchak konturi; 2)
:
sin
sinusoidaning ) 0
( O nuqtadan ) 0
0 (
nuqtagacha bo‘lgan yoyi. 4.5. 1)
yxdl , x y L 6 : 2 parabolaning y x 6 2 parabola kesgan yoyi; 2) L xdy ydx ,
r L : aylananing musbat yo‘nalishda aylanib o‘tishdagi yoyi.
4.6. 1)
dl y 2 , ) cos
1 ( 3 ), sin
( 3 : t y t t x L sikloidaning bir arkasi; 2) L xdz zdx sin
cos , : L
) 2 ; 0 ; 2 ( A va
) 2 ; 0 ; 2 ( B nuqtalarni tutashtiruvchi AB to‘g‘ri chiziq kesmasi. 4.7. 1)
xydl , : L tomonlari 1 , 1 , 1 , 1
y x x bo‘lgan kvadrat konturi; 2)
L y x xdy ydx 2 2 , :
)
; 1 ( A va
) 6 ; 3 (
nuqtalarni tutashtiruvchi
to‘g‘ri chiziq kesmasi.
L dl y x 2 2 , y y x L 2 : 2 2 aylana; 2)
dy y x dx y x ) ( ) ( 2 2 ,
: L
siniq chiziq, ), 0 ; 2 ( A ). 0 ; 5 ( ), 3 ; 5 (
B
dl y x ) ( , : L 4 4 2 cos 2 r Bernulli limniskatasining bo‘lagi; 2)
xdy y ydx x 2 cos sin 4 2 , :
)
; 0 ( A va
) 6 ; 3 (
nuqtalarni tutashtiruvchi
to‘g‘ri chiziq kesmasi. 23
4.10. 1) dl y x L ) 3 4 ( 3 , : L
) 0 ; 1 ( A va
) 1 ; 0 (
nuqtalarni tutashtiruvchi to‘g‘ri chiziq kesmasi; 2) L y x dx y dy x 3 5 3 5 2 2 3 , : L
y t x 3 3 sin 2 , cos 2 astroidaning ) 0
2 (
nuqtadan ) 2 ; 0 (
nuqtagacha bo‘lgan yoyi.
) ( 2 2 , : L
2 r aylananing birinchi choragi; 2) L dy x y xydx ) ( , :
3
y kubik parabolaning ) 0
0 (
nuqtadan ) 1 ; 1 ( B nuqtagacha bo‘lgan yoyi. 4.12. 1)
ydl , 2 : x y L parabolaning ) 4 ; 2 (
va )
; 1 ( B nuqtalar orasidagi yoyi; 2) L xdy ydx , : L
2 0 sin , cos
3 3
t t a y t a x astroida yoyi. 4.13. 1) L dl y x ) ( , ax y x L 2 : 2 2 aylana; 2)
dy y x dx y x ) ( ) ( , : L t y t x sin
3 , cos 2 ellipsning musbat yo‘nalishda aylanib o‘tishdagi yoyi. 4.14. 1) L dl y z 2 2 , 4 : 2 2 y z L aylana; 2) L xdy xdx y sin
cos , : L uchlari ), 0
1 (
) 0
2 ( ), 2 ; 0 ( C B nuqtalarda bo‘lgan ABC uchburchakning musbat yo‘nalishda aylanib o‘tishdagi konturi. 4.15. 1) L dl z y x ) ( 2 2 2 , t z t y t x L 3 , sin 4 , cos 4 : vint chizig‘ining birinchi o‘rami; 2) L dx y x ) ( 2 , : L
2 , 1 , 0 , 0 y x y x to‘g‘ri chiziqlardan tuzilgan to‘g‘ri to‘rtburchakning musbat yo‘nalishda aylanib o‘tishdagi konturi.
ydl , t y t x L 3 3 sin , cos : astroidaning ) 0 ; 1 ( A va
) 1 ; 0 (
nuqtalar orasidagi yoyi; 2) L xdy dx y xy ) ( 2 ,
: L 2 2x y parabolaning ) 0
0 (
nuqtadan ) 2 ; 1 ( B nuqtagacha bo‘lgan yoyi. 24
4.17. 1) L y x dl ,
: L
) 4 ; 0 ( A va
) 0 ; 4 (
nuqtalarni tutashtiruvchi to‘g‘ri chiziq kesmasi; 2)
xdy , :
2 2
R y x aylananing musbat yo‘nalishda aylanib o‘tishdagi yoyi.
4.18. 1) L dl y x 2 2 , x y x L 2 : 2 2 aylana; 2) L x x dy e x dx xye ) 1 ( , : L
) 2 ; 0 ( A va
) 2 ; 1 (
nuqtalarni tutashtiruvchi
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