6
Задание 4. Вычислить площадь области (с помощью перехода к полярной системе координат)
4.1. x2 + y2 ≥1, x2 + y2 ≤9 , y ≥ |
x |
|
, y ≤ x |
|
. |
||
|
3 |
||||||
|
|
|
|||||
3 |
|||||||
|
|
|
|
|
|||
4.2.x2 + y2 ≥ 2 , x2 + y2 ≤ 4 , y ≥ x , x ≥ 0 .
4.3.x2 + y2 ≤ 4 , y ≥ x , y ≤ x
3 .
4.4. x2 + y2 ≤1, y ≥ |
x |
|
, y ≤ x |
|
, x ≥ 0 . |
||
|
3 |
||||||
|
|
|
|||||
3 |
|||||||
|
|
|
|
|
|||
4.5.x2 + y2 ≥1, x2 + y2 ≤ 4 , −1 ≤ x ≤ 0 , 0 ≤ y ≤1.
4.6.x2 + y2 ≤ R2 – первая четверть круга.
4.7.x2 + y2 ≤9, y ≤ x , y ≥ −x .
4.8. x2 + y2 ≤ 4 , y ≥ |
x |
|
, y ≤ x |
|
, x > 0 . |
||
|
3 |
||||||
|
|
|
|||||
3 |
|||||||
|
|
|
|
|
|||
4.9. x2 + y2 ≥1, x ≥ 0 , x2 + y2 ≤ 4 , y ≥ 0 .
4.10. x2 + y2 ≥ |
π2 |
, |
x2 + y2 ≤π . |
|
4 |
||||
|
|
|
4.11.x2 + y2 ≤9 , y ≥ x , x ≥ 0 .
4.12.x2 + y2 ≥ a2 , x2 + y2 ≤ 4a2 .
4.13. x2 + y2 ≥ |
π2 |
, |
x2 + y2 ≤π2 . |
|
9 |
||||
|
|
|
4.14.y ≤ 
1− x2 , y ≥ 0 .
4.15.x2 + y2 ≤π2 .
4.16.x2 + y2 ≥ e2 , x2 + y2 ≤ e4 .
4.17.x2 + y2 ≤ a2 , x ≥ 0 , y ≥ 0 .
4.18.x2 + y2 ≤1, −1 ≤ x ≤ 0 , 0 ≤ y ≤1.
4.19.x2 + y2 ≤ 4x .
|
|
|
|
|
|
|
|
7 |
|
|
|
4.20. x2 |
+ y2 |
≤1, |
y ≥ 0 . |
|
|
|
|
|
|||
4.21. x2 |
+ y2 |
≤1, |
y ≥ |
x |
|
, |
y ≤ x |
|
, x > 0 . |
||
|
3 |
||||||||||
|
|
|
|||||||||
3 |
|||||||||||
|
|
|
|
|
|
|
|
|
|||
4.22.x2 + y2 ≤ 4 , y = x , x ≥ 0 .
4.23.x2 + y2 ≥1, x2 + y2 ≤ 4 .
4.24.π2 ≤ x2 + y2 ≤ 4π2 .
4.25.x2 + y2 ≤ 9, y = x , y ≥ 0 .
4.26. x2 + y2 ≥1, x2 + y2 ≤ 9, y ≥ |
x |
|
, y ≤ x |
|
. |
|||||
|
3 |
|||||||||
|
|
|
||||||||
3 |
||||||||||
|
|
|
|
|
|
|
|
|||
4.27. x2 |
+ y2 |
≤ Rx . |
|
|
|
|
|
|
|
|
4.28. x2 |
+ y2 |
≤ 4x , |
y ≤ x . |
|
|
|
|
|
|
|
4.29. x2 |
+ y2 |
≤1, x ≥ 0 , y ≥ 0 . |
|
|
|
|
|
|
||
4.30. x2 |
+ y2 |
≤ 2y , |
y ≤ x . |
|
|
|
|
|
|
|
Задание 5. Вычислить тройной интеграл.
5.1. ∫∫∫dxdydz |
; V : |
z2 ≤ x2 + y2 , z ≥1, z ≤ 4 . |
|
V |
z |
|
|
dxdydz
5.2. ∫∫∫V (1+ x + y + z)3 ; V : x = 0 , y = 0, z = 0 , x + y + z =1.
5.3. ∫∫∫xy2 z3dxdydz ; V : z = xy , y = x , x =1, z = 0 .
V
5.4. ∫∫∫xyzdxdydz ; V : x2 + y2 + z2 =1, x = 0 , y = 0 , z = 0 .
V |
|
|
|
|
|
5.5. ∫∫∫dxdydz |
; V : |
x2 |
= y2 + z2 , |
x =1, x = 4 . |
|
V |
x |
|
|
|
|
5.6. ∫∫∫y2dxdydz ; V : |
y ≤ x2 + z2 , |
y ≤ 2, y ≥ 6 . |
|||
V
8
5.7. ∫∫∫x3dxdydz ; V : x ≤ 4 − y2 − z2 , x ≥ 0 , x ≤1.
V |
|
|
; V : (y2 −1)+ z2 |
|
|
|
|
5.8. ∫∫∫dxdydz |
|
≤1, x ≥ 2 , x ≤ 4 . |
|||||
V |
x |
|
|
|
|
|
|
5.9. ∫∫∫ |
dxdydz |
; V : x2 + y2 + z2 |
≤16 , |
y ≥ 2, |
y ≤3 . |
||
y |
2 |
||||||
V |
|
|
|
|
|
|
|
5.10. ∫∫∫(x + 2y + z)dxdydz ; V : x = 0 , |
y = 0, |
z = 0 , x =1, |
|||||
V |
|
|
|
y + z = 2 . |
|
||
|
|
|
|
|
|||
5.11. ∫∫∫zdxdydz ; V : 0 ≤ x ≤ 1 |
, |
x ≤ y ≤ 2x , |
|
||||
V |
|
|
2 |
|
|
|
|
0 ≤ z ≤ 
1− x2 − y2 .
5.12. ∫∫∫
x2 + y2 dxdydz ; V : x2 + y2 ≤ z2 , z ≤1.
V
5.13. ∫∫∫xdxdydz ; V : x ≥ 0 , |
y ≥ 0 , z ≥ 0 , y ≤ h , x + z ≤ a . |
|||||
V |
|
|
|
|
|
|
5.14. ∫∫∫dxdydz |
; V : z ≥ 0 , |
z ≤1, x ≥ 0 , y ≥ 0 , x + y ≤1. |
||||
V |
x |
|
|
|
|
|
5.15. ∫∫∫ |
dxdydz |
; V : x2 + z2 + z2 ≤9 , |
x ≤1, |
z ≤ 2 . |
||
z |
2 |
|||||
V |
|
|
|
|
|
|
5.16. ∫∫∫dxdydz |
; V : (x −1)2 + y2 ≤1, |
z ≥3 , |
z ≤ 6 . |
|||
V |
z |
|
|
|
|
|
5.17. ∫∫∫dxdydz |
; V : z ≤ y − x2 − y2 , |
z ≥ 0 , |
z ≤3 . |
|||
V |
x |
|
|
|
|
|
5.18. ∫∫∫z2dxdydz ; V : z ≤ x2 + y2 , z ≥ 2 , z ≤ 6 . |
||||||
V |
|
|
|
|
|
|
5.19. ∫∫∫dxdydz |
; V : z ≤ x2 + y2 , z ≥1, z ≤ 4 . |
|||||
V |
z |
|
|
|
|
|
9
5.20. ∫∫∫(x2 + y2 + z2 )dxdydz ; V : x ≥ 0 , y ≥ 0 , z ≥ 0 ,
V
x ≤ a > 0 , y ≤ b > 0 , z ≤ c > 0 .
5.21. ∫∫∫ydxdydz ; V : x ≥ 0 , x ≤ 2 , y ≥ 0 , y ≤1, z ≤ 0 ,
V
z ≤1− y .
5.22. ∫∫∫xz2dxdydz ; V : x = 
2y − y2 , x ≤ 2 , y ≥ 0 , y ≤ 2,
V
z ≥ 0 , z ≤ 3 . |
|
5.23. ∫∫∫(2x +3y − z)dxdydz ; V : x ≥ 0 , |
y ≥ 0 , x + y ≤3 , |
V |
|
z ≥ 0 , |
z ≤ 4 . |
5.24. ∫∫∫zdxdydz ; V : z ≥ 
x2 + y2 , 2z ≤8 − x2 − y2 .
V
5.25. ∫∫∫xyzdxdydz ; V : x ≥ 0 , y ≥ 0 , z ≥ 0 , x + y + z ≤1.
V |
|
|
|
|
5.26. ∫∫∫(1+ x)dxdydz ; V : x ≥ 0 , |
y ≥ 0 , |
z ≥ 0 , y ≤1, |
||
V |
|
|
|
|
y ≤1− x , x + y + z ≤1. |
||||
5.27. ∫∫∫xy2 z3dxdydz ; V : x ≤1, |
y ≤ x , z ≥ 0 , z ≤ xy . |
|||
V |
|
|
|
|
5.28. ∫∫∫y cos(x + z)dxdydz ; V : y ≤ |
|
, |
y ≥ 0 , z ≥ 0 , |
|
x |
||||
V |
|
|
|
|
x + z ≤ π2 .
5.29. ∫∫∫xydxdydz ; V : z = xy , x + y ≤1, y ≥ 0 , z ≥ 0 .
V
5.30. ∫∫∫
x2 + y2 dxdydz ; V : x2 + y2 = z2 , z =1.
V