Задание 6. Вычислить объём тела, ограниченного поверхностями.

x = 4y2 ,

6.4.8x + 8y + 4z =1,z = 0,

y = 0.

6.5.z =1− x2 − 4y2 ,z = 0.

x + y + z = 2,y = x2 ,

6.19.

z = 0,x = 0.

z = x2 − y2 ,

x =1,

6.20.

y = 0,z = 0.

x2 + y2 = 9,z = 5x,

6.6.

z = 0,

y = 0 (y ≥ 0).

z = 9 − y2 ,z = 0,

6.7. x = 0,

y = 0 (y ≥ 0),

3x + 4y =12.

z = x2 + y2 ,x = 0,

6.8. z = 0,

y =1,y = x2 .

z = 9 − x2 ,y = 2x,

6.9.

y = 0,z = 0.

z = 9 − x2 ,

z = 0,

6.10. y = x2 ,

y = 0.

11

z = 4 − y2 ,x2

6.21. y = 2 ,

z = 0,x = 0.

z = 4 − x2 ,2x + y = 4,

6.22. x = 0 (x ≥ 0),

y = 0,

z = 0.

z =16 − x2 ,y = x,

6.24.

y = 0,z = 0.

z = 4 − y2 ,

z = 0,

6.25. x = y2 ,

x = 0.

z =1− x2 − y2 ,

z = 0,

6.13.x = 0,y = 0,

x + y =1.

z = 9 − x2 − y2 ,

x + y = 3,

6.14.z = 0,x = 0,y = 0.

x2 + y2 = 4,

6.15.z = y,

x = 0 (x ≥ 0).

12

z = 4 − x2 − y2 ,

z = 0,

6.28.x = 0,y = 0,

x + y = 2.

z =16 − x2 − y2 ,

x + y = 2,

6.29.x = 0,y = 0,z = 0.

z =1− y2 ,x2

6.30. y = 4 ,

z = 0,x = 0.

7.1. D : 2x ≤ x2 + y2 ≤ 4x,

0 ≤ y ≤ x,

µ= 2y .

µ= 3y .

µ=16y .

µ=12xy .

µ= xy .

µ= xy .

µ= arctg xy .

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1 ≤ x2 + y2 ≤ 4,

7.8. D : y ≥ 0,

y ≤ x3,

π 2 ≤ x2 + y2 ≤ 4π 2 ,

7.9. D : y ≥ 0,

y ≥ −x,

1 ≤ x2 + y2 ≤ 4,

7.10. D : y ≥ 0,

x ≥ 0,

2x ≤ x2 + y2 ≤ 4x,

7.14. D : y ≥ 0,

x ≥ 0,

2y ≤ x2 + y2 ≤ 4y,

7.16. D : y ≥ 0,

x ≤ 0,

µ= x2 + y2 .

µ= sin x2 + y2 .

µ= 4 − x2 − y2 .

µ= e−(x2 +y2 ).

µ= 1− x2 − y2 .

15

7.17.D : e2 ≤ x2 + y2 ≤ e4 ,

y ≥ 0,

µ= ln(x2 + y2 ).

µ=π2 − y2 .

x2

µ = sin x2 + y2 . x2 + y2

1

µ = x2 + y2 +1 .

µ = x2 + y2 .

µ = xy .

16

µ= xy .

µ= arctg xy .

µ=1+3x +3y .

µ= x .

µ= xy .

µ= x2 . y2