10

Задание 6. Вычислить объём тела, ограниченного поверхностями.

2y2 = x,

x + y + z = 3,

x y z

x2 +y2

=1,

+

+

=1,

6.1.

4

2

4

6.16.

x = 0,

z = 0,

y = 0.

y = 0,

z = 0.

x + y − z +1 = 0,

x + y + z =1,

=

0,

6.2.

z

6.17.

xy = z,

x =

0,

2

.

z = 0.

y = x

x + y = 3,

3x +3y + z = 0,

= x

2

+ y

2

,

z

6.3.

0,

6.18.

x =1,

x =

y

=

0,

y = 0,

0.

z = 0.

z =

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.

			y2
	z	=		,
			2
6.23.
	2x +3y −12 = 0,
	x	= 0,
				z = 0,
	y = 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 ,
	6.11.	x + y + z =1,
			(x ≥ 0),
		x = 0
			(y ≥ 0).
		y = 0
		z = 0 − x2 − y2 ,
				z
	6.12.	x + y	+		= 3,
			3
		x = 0	(x ≥ 0),
			(y ≥ 0).
		y = 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
	6.26.	x + y	+					= 2,
			2					0),
		x = 0	(x ≥
			(y ≥ 0).
		y = 0
		z =16 − x2 − y2 ,
					z
	6.27.	x + y	+					= 4,
			4					0),
		x = 0	(x ≥
			(y ≥ 0).
		y = 0

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.

13

Задание 7. Найти массу плоскости пластины D , с помощью перехода к ПСК.

7.1. D : 2x ≤ x2 + y2 ≤ 4x,

0 ≤ y ≤ x,

	x ≤ x2 + y2 ≤ 2x,
		x
	7.2. D :				≤ y ≤ x,
		3

y ≤ x2							+ y2 ≤ 2y,
		0,
7.3. D : x ≥
y ≥ x
							3,
y ≤ x2							+ y2 ≤ 3y,
		0,
7.4. D : y ≥
y ≤ x,
y ≤ x2							+ y2 ≤ 3y,
7.5. D : y ≤ x,
y ≥ 0,
− y		≤ x2 + y2 ≤ −2y,
			x
7.6. D : y ≤							,
		3
							3,
y ≥ x
x2 + y2								≥1,
								≤ 9,
x2 + y						2
7.7. D :	y ≥						x				,
							3
	y ≤						3x,

µ= 2y .

µ= 3y .

µ=16y .

µ=12xy .

µ= xy .

µ= xy .

µ= arctg xy .

14

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,

4 ≤ x2	+ y2	≤ 9,
7.11. D :
y ≥ 0,

x2 + y2					≤1,
	y ≥		x				,
7.12. D :			3
	y ≤ x
					3,
	x ≥		0,
4 ≤ x2			+ y2 ≤ 9,
7.13. D :
y ≥ 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 .

µ =				1				.
		x2		+ y2	+1
µ = x2 + y2 .
µ =				1			.
	x2			+ y2	+1

15

7.17.D : e2 ≤ x2 + y2 ≤ e4 ,

y ≥ 0,

4 ≤ x2 + y2 ≤ π 2 ,

0,

7.18. D : y ≥

x ≤ 0,

2 + y2

≥ π

2

,

x

2

2

9

+ y

≤ π

2

,

7.19. D : x

y ≤ 0,

y ≥ x,

x2 + y2

≤ y,

0,

7.20. D : x ≤

y ≤ 0,

− 2y ≤ x2 + y2 ≤ −4y,

0,

7.21. D : y ≤

x ≥ 0,

4 ≤ x2 + y2

≤ 9,

7.22. D : y ≥ x,

x ≥ 0,

π 2

≤ x2

+ y2

≤

π

2

,

4

36

x

7.23. D : y ≤

,

3

y ≥

3

x,

µ= ln(x2 + y2 ).

µ=π2 − y2 .

x2

µ = sin x2 + y2 . x2 + y2

1

µ = x2 + y2 +1 .

µ = x2 + y2 .

µ = xy .

µ =	cos		x2 + y2				.
		x2		+ y2

16

1 ≤ x2 + y2									≤ 4,
		x
7.24. D : y ≤							,
	3
y ≥	0,
x2 + y2								≤ y,
		x
7.25. D : y ≥							,
	3
y ≤ x 3,
x2 + y2								≤ 9,
	0
7.26. D : y ≥
x ≥ 0
1 ≤ x2 + y2									≤ 4,
	0,
7.27. D : y ≤
x ≥	0,
4 ≤ x2 + y2									≤ 9,
		x
7.28. D : y ≥							,
	3
x ≥	0,
1 ≤ x2 + y2									≤ 4,
		x
7.29. D : y ≥							,
	3
x ≥	0,
− y ≤ x2 + y2 ≤ −2y,
	0,
7.30. D : x ≥
y ≤ −x,

µ= xy .

µ= arctg xy .

µ=1+3x +3y .

µ =	1		.
		1+ x2 + y2

µ= x .

µ= xy .

µ= x2 . y2