Задача 3. Вычислить.

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çæ	6x2 y2	+	25		x4 y4	÷ödxdy;

2.25. òòD è			3			ø
D : x =1, y = x2 , y = -								.
							x
çæ	3x2 y2	+	50	x4 y4		÷ödxdy;

2.27. òòD è			3			ø

D : x =1, y = 3x, y = -x3.

òò(54x2 y2 +150x4 y4 )dxdy;

2.29.D

D : x =1, y = x2 , y = -3x.

òò(54x2 y2 +150x4 y4 )dxdy;

2.31.D

D : x =1, y = x3 , y = -x.

y = 9x, y = 0, x =1,

3.1. òòD yexy / 2dxdy;

D : y = ln 2, y = ln3, x = 2, x = 4.

3.3. òòD y cos xy dxdy;

D : y = π / 2, y = π , x =1, x = 2.

3.5. òòD ysin xy dxdy;

D : y = π / 2, y = π , x =1, x = 2.

òò4ye2xydxdy;

3.7. D

D : y = ln3, y = ln 4, x = 12 , x =1.

òò(9x2 y2 + 25x4 y4 )dxdy;

2.26.D

D : x =1, y = x, y = -x2.

òò(9x2 y2 + 25x4 y4 )dxdy;

2.28.D

D : x =1, y = x3, y = -3x.

òò(xy - 9x5 y5 )dxdy;

2.30.D

D : x =1, y = 3x, y = -x2.

	òò y2 sin	xy	dxdy;
	òò y2 sin	2	dxdy;
3.2.	D	2
3.2.	D					x
	D : x = 0, y =				, y =		.
				π
				π
				2
3.4.	òò y2e− xy / 4dxdy;
3.4.	D
	D : x = 0, y = 2, y = x.
	òò y2 cos xy dxdy;
3.6.	D	2
	D : x = 0, y =
	D : x = 0, y =			π 2, y = x 2.
	òò4y2 sin xy dxdy;
3.8.	D
3.8.	D : x = 0, y =			π , y = x.
	D : x = 0, y =			π , y = x.
				2

161

òò y cos2xy dxdy;

3.9. D

D : y = π2 , y = π , x = 12 , x =1.

òò12ysin 2xy dxdy;

3.11. D

D : y = π4 , y = π2 , x = 2, x = 3.

3.13. òòD yexy / 4dxdy;

D : y = ln 2, y = ln3, x = 4, x = 8.

òò2y cos2xy dxdy;

3.15. D

D : y = π4 , y = π2 , x =1, x = 2.

òòysin xy dxdy;

3.17.D

D : y = π , y = 2π , x = 12 , x =1.

òò8ye4xydxdy;

3.19.D

D : y = ln3, y = ln 4, x = 14 , x = 12.

òòy cos xy dxdy;

3.21.D

D : y = π , y = 3π , x =12, x =1.

òòysin 2xy dxdy;

3.23.D

D : y = π2, y = 3π2, x =12, x = 2.

òòy2e−xy /8dxdy;

3.10.D

D : x = 0, y = 2, y = 2x .

òòy2 cos xy dxdy;

3.12.D

D : x = 0, y = π , y = x.

òòy2 sin 2xy dxdy;

3.14.D

D : x = 0, y = 2π , y = 2x.

òòy2e−xy / 2dxdy;

3.16.D

D : x = 0, y = 2, y = x.

òò y2 cos2xy dxdy;

3.18. D

D : x = 0, y = π2 , y = 2x .

òò3y2 sin 2 dxdy;

3.20. D

D : x = 0, y = 43π , y = 23 x.

òòy2e− xy / 2dxdy;

3.22.D

D : x = 0, y =1, y = 2x .

òòy2 cos xy dxdy;

3.24.D

D : x = 0, y = π , y = 2x.

162

3.25. òòD 6yexy /3dxdy;

D : y = ln 2, y = ln3, x = 3, x = 6.

òòy cos2xy dxdy;

3.27.D

D : y = π2, y = 3π2, x =12, x = 2.

òò3ysin xy dxdy;

3.29.D

D : y = π2, y = 3π, x =1, x = 3.

òò12ye6 xy dxdy;

3.31. D

D : y = ln3, y = ln 4, x =16, x =13.

	òò y2 sin	xy	dxdy;
3.26.	D	2

D : x = 0, y = π , y = x.

3.28.	òò y2e− xy /8dxdy;
	D
	D : x = 0, y = 4, y = 2x.
	òò y2 cos xy dxdy;
3.30.	D	2

D : x = 0, y = 2π , y = 2x.

Задача 4. Вычислить.

òòò2y2exy dx dy dz;

	V
	4.1.	ìx = 0, y =1, y = x,
	V	ìx = 0, y =1, y = x,
	V	í
		îz = 0, z =1.

òòò y2ch(2xy) dx dy dz;

4.3. ìx =

V íîz =

òòòx2sh

0, y = -2, y = 4x, 0, z = 2.

(3xy) dx dy dz;

	V
	4.5.	ìx =1, y = 2x, y = 0,
	V	ìx =1, y = 2x, y = 0,
	V	í
		îz = 0, z = 36.

òòòx2 zsin(xyz) dx dy dz;

V
4.2.	ìx	= 2, y = π , z =1,
V	ìx	= 2, y = π , z =1,
V	í	= 0, y =1, z = 0.
	îx	= 0, y =1, z = 0.
òòò8y2 z e2xyz dx dy dz;
V
4.4.	ìx	= -1, y = 2, z =1,
V	ìx	= -1, y = 2, z =1,
V	í	= 0, y = 0, z = 0.
	îx	= 0, y = 0, z = 0.
òòò y2 z cos(xyz) dx dy dz;
V
4.6.	ìx	=1, y = 2π , z = 2,
V	ìx	=1, y = 2π , z = 2,
V	í	= 0, y =1, z = 0.
	îx	= 0, y =1, z = 0.

163

	2	æ π	ö
òòò y		cosç	xy ÷ dx dy dz;
4.7. V		è 4	ø
4.7. V	ìx = 0, y = -1, y = x 2,
V	ìx = 0, y = -1, y = x 2,
V	í	= 0, z	= -π 2.
	îz	= 0, z	= -π 2.
òòò y2e− xy dx dy dz;
V
4.9.	ìx	= 0, y	= -2, y = 4x,
V	ìx	= 0, y	= -2, y = 4x,
V	í

z=1.

òòòy2ch(2xy) dx dy dz;îz = 0,

V
4.11.	ìx	= 0, y =1, y = x,
V	ìx	= 0, y =1, y = x,
V	í	= 0, z = 8.
	îz	= 0, z = 8.
òòò y2exy 2			dx dy dz;
V
4.13.	ìx	= 0, y = 2, y = 2x,
V	ìx	= 0, y = 2, y = 2x,
V	í	= 0, z = -1.
	îz	= 0, z = -1.
	2	æ	π xy ö
òòò y		cosç	÷ dx dy dz;
4.15. V		è	2 ø
4.15. V	ìx = 0, y = -1, y = x,
V	ìx = 0, y = -1, y = x,
V	í	= 0, z = 2π 2.
	îz	= 0, z = 2π 2.
òòò y2cos(π xy) dx dy dz;
V
4.17.	ìx	= 0, y =1, y = 2x,
V	ìx	= 0, y =1, y = 2x,
V	í

z= π 2.

òòòx2sh (2xy) dx dy dz;îz = 0,

	V
	4.19.	ìx = -1, y = x, y = 0,
	V	ìx = -1, y = x, y = 0,
	V	í
		îz = 0, z = 8.

òòòx2 zsin				xyz		dx dy dz;
òòòx2 zsin						dx dy dz;
4.8.	V		4
4.8.	V	ìx =1, y = 2π , z = 4,
V		ìx =1, y = 2π , z = 4,
V		í
		îx = 0, y = 0, z = 0.
	òòò2y2 z e2xyz dx dy dz;
4.10.	V
4.10.		ìx	=1, y =1, z =1,
	V	ìx	=1, y =1, z =1,
	V	í	= 0, y = 0, z = 0.
		îx	= 0, y = 0, z = 0.
	òòòx2 z sh (xyz) dx dy dz;
4.12.	V
4.12.		ìx	= 2, y =1, z =1,
	V	ìx	= 2, y =1, z =1,
	V	í	= 0, y = 0, z = 0.
		îx	= 0, y = 0, z = 0.
	òòò y2 z cos				xyz		dx dy dz;
	òòò y2 z cos						dx dy dz;
4.14.	V		3
	V	ìx = 3, y =1, z = 2π ,
	V	í	= 0, y = 0, z = 0.
		îx	= 0, y = 0, z = 0.
	òòòx2 z sh (xyz) dx dy dz;
4.16.	V
4.16.		ìx	=1, y = -1, z =1,
	V	ìx	=1, y = -1, z =1,
	V	í	= 0, y = 0, z = 0.
		îx	= 0, y = 0, z = 0.
	òòò2x2 z sh(2xyz) dx dy dz;
4.18.	V
4.18.		ìx	= 2, y =1 2, z =1 2,
	V	ìx	= 2, y =1 2, z =1 2,
	V	í	= 0, y = 0, z = 0.
		îx	= 0, y = 0, z = 0.
	òòòx2 z sin				xyz		dx dy dz;
	òòòx2 z sin						dx dy dz;
4.20.	V		2
	V	ìx =1, y = 4, z = π ,
	V	í	= 0, y = 0, z = 0.
		îx	= 0, y = 0, z = 0.

164

òòò y2ch(xy) dx dy dz;

V
4.21.	ìx	= 0, y = -1, y = x,
V	ìx	= 0, y = -1, y = x,
V	í	= 0, z = 2.
	îz	= 0, z = 2.
	2	æ	π	xy	ö
òòòx		cosç	2	xy	÷ dx dy dz;
4.23. V		è	2		ø
4.23. V	ìx = 2, y = x, y = 0,
V	ìx = 2, y = x, y = 0,
V	í	= 0,	z = π.
	îz	= 0,	z = π.
òòòx2cos(π xy)					dx dy dz;
V
4.25.	ìx	=1, y = 2x, y = 0,
V	ìx	=1, y = 2x, y = 0,
V	í	= 0, z = 4π.
	îz	= 0, z = 4π.
òòò y2ch(3xy) dx dy dz;
V
4.27.	ìx	= 0, y = 2, y = 6x,
V	ìx	= 0, y = 2, y = 6x,
V	í	= 0, z = -3.
	îz	= 0, z = -3.
òòòx2sin(4π xy) dx dy dz;
V
4.29.	ìx	=1, y = x 2, y = 0,
V	ìx	=1, y = x 2, y = 0,
V	í	= 0, z = 8π.
	îz	= 0, z = 8π.
òòòx2sh(xy) dx dy dz;
V
4.31.	ìx	= 2, y = x 2, y = 0,
V	ìx	= 2, y = x 2, y = 0,
V	í	= 0, z =1.
	îz	= 0, z =1.

Задача 5. Вычислить.

	òòòx2 z ch(xyz)								dx dy dz;
4.22.	V
4.22.		ìx		=1, y =1, z =1,
	V	ìx		=1, y =1, z =1,
	V	í		= 0, y = 0, z = 0.
		îx		= 0, y = 0, z = 0.
	òòò y2 z cos						xyz	dx dy dz;
	òòò y2 z cos							dx dy dz;
4.24.	V					9
4.24.	V	ìx = 9, y =1, z = 2π ,
	V	ìx = 9, y =1, z = 2π ,
	V	í		= 0, y = 0, z = 0.
		îx		= 0, y = 0, z = 0.
	òòò y		2			æ xyz			ö
	òòò y			z chç					÷ dx dy dz;
4.26.	V					è 2			ø
4.26.	V	ìx = 2, y = -1, z = 2,
	V	ìx = 2, y = -1, z = 2,
	V	í		= 0, y = 0, z = 0.
		îx		= 0, y = 0, z = 0.
	òòò2y2 z ch(2xyz) dx dy dz;
	V
4.28.		ì			1
	V	ïx		=		, y = 2, z = -1,
		ïx		=	2	, y = 2, z = -1,
		í			2
		ï		= 0, y = 0, z = 0.
		îx		= 0, y = 0, z = 0.
	òòò8y2 z e−xyz dx dy dz;
4.30.	V
4.30.		ìx		= 2, y = -1, z = 2,
	V	ìx		= 2, y = -1, z = 2,
	V	í		= 0, y = 0, z = 0.
		îx		= 0, y = 0, z = 0.

165

òòòx dx dy dz;

5.1. V : y =10x, y = 0, x =1, z = xy, z = 0.

òòò15(y2 + z2 ) dx dy dz;

5.3.V : z = x + y, x + y =1,

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

òòò(1+ 2x3 ) dx dy dz;

5.5. V : y = 9x, y = 0, x =1, z = xy, z = 0.

òòò y dx dy dz;

5.7. V : y =15x, y = 0, x =1, z = xy, z = 0.

òòò(3x2 + y2 ) dx dy dz;

5.9.V : z =10y, x + y =1,

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

òòò(4 + 8z3 ) dx dy dz;

5.11. V : y = x, y = 0, x =1, z = xy, z = 0.

òòòV			dx dy dz						;
	æ	+	x	+	y	+	z ö4
	ç1							÷
			3		4		8
	è							ø

5.2.V : 1+ 3x + 4y + 8z =1,

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

òòò(3x + 4y) dx dy dz;

5.4. V : y = x, y = 0, x =1,

z = 5(x2 + y2 ), z = 0.

òòò(27 + 54y3 ) dx dy dz;

5.6. V : y = x, y = 0,

x =1,

z =

, z = 0.

òòòV

dx dy dz

;

ö5

ç1

5.8.V : 16x + 5y + 3z =1,

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

òòò(15x + 30z) dx dy dz;

5.10. V : z = x2 + 3y2 , z = 0, y = x, y = 0, z = 0.

òòò(1+ 2x3 ) dx dy dz;

5.12. V : y = 36x, y = 0, x =1, z = xy, z = 0.

166

òòò21xz dx dy dz;

5.13. V : y = x, y = 0, x = 2, z = xy, z = 0.

òòò(x2 + 3y2 ) dx dy dz;

5.15.V : z =10x, x + y =1,

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

òòò

5.17. V :

òòò

3y2 dx dy dz;

5.19. V : y = 2x, y = 0, x = 2, z = xy, z = 0.

òòòx2 dx dy dz;

5.21.V : z =10(x + 3y), x + y =1,

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

òòò63(1+ 2 y ) dx dy dz;

5.23. V : y = x, y = 0, x =1, z = xy, z = 0.

òòòV

dx dy dz

;

ö6

ç1

5.14. V :

=1,

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

òòò(60y + 90z) dx dy dz;

5.16. V : y = x, y = 0, x =1,

z = x2 + y2 , z = 0.

òòò(9 +18z) dx dy dz;

5.18. V : y = 4x, y = 0, x =1,

z =

, z = 0.

òòòV

dx dy dz

;

ö6

ç1+

5.20. V :

=1,

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

òòò(8y +12z) dx dy dz;

5.22. V : y = x, y = 0, x =1,

z = 3x2 + 2y2 , z = 0.

òòò(x + y) dx dy dz;

5.24. V : y = x, y = 0, x =1,

z = 30x2 + 60y2 , z = 0.

167

òòòV			dx dy dz								;
	æ	+		x	+	y	+		z	ö6
	ç1									÷
			6			4		16
	è									ø

5.25.V : 6x + 4y + 16z =1,

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

òòò y2 dx dy dz;

5.27.V : z =10(3x + y), x + y =1,

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

òòò(x2 + 4y2 ) dx dy dz;

5.29.V : z = 20(2x + y), x + y =1,

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

òòòx2 z dx dy dz;

òòòxyz dx dy dz;

5.26. V : y = x, y = 0, x = 2, z = xy, z = 0.

òòòV	æ	5x +	3z ö	dx dy dz;
	è		2 ø
	ç		÷

5.28. V : y = x, y = 0, x = 2,

z = x2 +15y2 , z = 0.

òòòV

dx dy dz

;

ö6

ç1+

5.30. V :

=1,

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

5.31. V : y = 3x, y = 0, x = 2, z = xy, z = 0.

Задача 6. Найти площадь фигуры, ограниченной данными линиями.

6.1.y = 3x, y = 4ex , y = 3, y = 4.

6.2.x = 36 - y2 , x = 6 - 36 - y2 .

6.3.x2 + y2 = 72, 6y = -x2 ( y £ 0).

6.4.x = 8 - y2 , x = -2y.

168

6.5. y = 3x , y = 8ex , y = 3, y = 8.

6.6. y =		x	, y =	1	, x =16.
	2			2x

6.7. x = 5 - y2 , x = -4y.

6.8. x2 + y2 = 12, -6y = x2 ( y £ 0).

6.9.y = 12 - x2 , y = 23 - 12 - x2 , x = 0 (x ³ 0).

6.10.y = 32 x, y = 23x , x = 9.

6.11.y = 24 - x2 , 23y = x2 , x = 0 (x ³ 0).

6.12.y = sin x, y = cos x, x = 0, (x ³ 0).

6.13.y = 20 - x2 , y = -8x.

6.14.y = 18 - x2 , y = 32 - 18 - x2 .

6.15.y = 32 - x2 , y = -4x.

6.16.y = 2x, y = 5ex , y = 2, y = 5.

6.17. x2 + y2 = 36, 32y = x2 ( y ³ 0).

6.18.y = 3x, y = 3x, x = 4.

6.19.y = 6 - 36 - x2 , y = 36 - x2 , x = 0 (x ³ 0).

6.20.y = 254 - x2 , y = x - 52.

6.21.y = x, y =1x, x =16.

169

6.22.y = 2x, y = 7ex , y = 2, y = 7.

6.23.x = 27 - y2 , x = -6y.

6.24.x = 72 - y2 , 6x = y2 , y = 0 ( y ³ 0).

6.25.y = 6 - x2 , y = 6 - 6 - x2 .

6.26.y = 32 x, y = 23x , x = 4.

6.27.y = sin x, y = cos x, x = 0, (x £ 0).

6.28.y = 1x , y = 6ex , y =1, y = 6.

6.29.y = 3x, y = 3x, x = 9.

6.30.y =11- x2 , y = -10x.

6.31.x2 + y2 = 12, x6 = y2 (x ³ 0).

Задача 7. Найти площадь фигуры, ограниченной данными линиями.


y2 - 2y + x2		= 0,	x2 - 4x + y2 = 0,
7.1. y2 - 4y + x2		= 0,	7.2. x2 -8x + y2	= 0,
y = x			y = 0, y = x
	3, y = 3x.				3.
y2 - 6y + x2		= 0,	x2 - 2x + y2	= 0,
7.3. y2 - 8y + x2		= 0,	7.4. x2 - 4x + y2	= 0,
y = x			y = 0, y = x.
	3, y = 3x.

170

y2 − 8y + x2 = 0,

7.5.y2 −10y + x2 = 0, y = x3, y = 3x.

y2 − 4y + x2 = 0,

7.7.y2 − 6y + x2 = 0, y = x, x = 0.

y2 − 6y + x2 = 0,

7.9.y2 −10y + x2 = 0, y = x, x = 0.

y2 − 2y + x2 = 0,

7.11.y2 − 4y + x2 = 0, y = 3x, x = 0.

y2 − 4y + x2 = 0,

7.13.y2 − 6y + x2 = 0, y = 3x, x = 0.

y2 − 2y + x2 = 0,

7.15.y2 − 6y + x2 = 0, y = x3, y = 0.

y2 − 2y + x2 = 0,

7.17.y2 −10y + x2 = 0, y = x3, y = 3x.

y2 − 4y + x2 = 0,

7.19.y2 −10y + x2 = 0, y = x3, y = 3x.

x2 − 4x + y2 = 0,

7.6.x2 −8x + y2 = 0, y = 0, y = x.

x2 − 2x + y2 = 0,

7.8.x2 −10x + y2 = 0, y = 0, y = 3x.

x2 − 2x + y2 = 0,

7.10.x2 − 4x + y2 = 0,

y = x3, y = 3x.

x2 − 2x + y2 = 0,

7.12.x2 − 6x + y2 = 0,

y = x3, y = 3x.

x2 − 2x + y2 = 0,

7.14.x2 −8x + y2 = 0,

y = x3, y = 3x.

x2 − 2x + y2 = 0,

7.16.x2 − 4x + y2 = 0, y = 0, y = x3.

x2 − 2x + y2 = 0,

7.18.x2 − 6x + y2 = 0, y = 0, y = x3.

x2 − 2x + y2 = 0,

7.20.x2 − 6x + y2 = 0, y = 0, y = x.

171

y2 - 2y + x2 = 0,

7.21.y2 - 4y + x2 = 0, y = x, x = 0.

y2 - 6y + x2 = 0,

7.23.y2 -8y + x2 = 0, y = x, x = 0.

y2 - 4y + x2 = 0,

7.25.y2 -8y + x2 = 0, y = x, x = 0.

y2 - 4y + x2 = 0,

7.27.y2 -8y + x2 = 0, y = 3x, x = 0.

y2 - 2y + x2 = 0,

7.29.y2 -10y + x2 = 0, y = x3, x = 0.

y2 - 4y + x2 = 0,

7.31.y2 - 8y + x2 = 0, y = x3, x = 0.

Задача 8. Пластинка

x2 - 2x + y2 = 0,

7.22.x2 - 4x + y2 = 0, y = 0, y = 3x.

x2 - 4x + y2 = 0,

7.24.x2 -8x + y2 = 0, y = 0, y = 3x.

x2 - 4x + y2 = 0,

7.26.x2 -8x + y2 = 0,

y = x3, y = 3x.

x2 - 4x + y2 = 0,

7.28.x2 - 6x + y2 = 0,

y = x3, y = 3x.

x2 - 6x + y2 = 0,

7.30.x2 -10x + y2 = 0, y = x3, y = 3x.

D задана ограничивающими ее кривыми, μ - поверхностная

плотность. Найти массу пластинки.
	D : x =1, y = 0, y2 = 4x ( y ³ 0);		D : x2 + y2 =1, x2 + y2 = 4,
8.1.	D : x =1, y = 0, y2 = 4x ( y ³ 0);	8.2.	x = 0, y = 0	(x ³ 0, y ³ 0);
	μ = 7x2 + y.		μ = (x + y)	(x2 + y2 ).
			μ = (x + y)	(x2 + y2 ).

172

D : x =1, y = 0, y2 = 4x		( y ³ 0);
8.3.	+ 5y.
μ = 7x2 2	+ 5y.

D : x = 2, y = 0, y2 = 2x		( y ³ 0);
8.5.	+ 2y.
μ = 7x2 8	+ 2y.

D : x = 2, y = 0, y2 = x 2		( y ³ 0);
8.7.	+ 6y.
μ = 7x2 2	+ 6y.

D : x =1, y = 0, y2	= 4x	( y ³ 0);
8.9.
μ = x + 3y2 .

D : x =1, y = 0, y2	= x	( y ³ 0);
8.11.
μ = 3x + 6y2 .

D : x = 2, y = 0, y2	= x 2	( y ³ 0);
8.13.
μ = 2x + 3y2 .

D : x =	1	, y = 0, y2	= 8x	( y ³ 0);
	2
8.15.

μ = 7x + 3y2.

D : x =1, y = 0, y2		= 4x	( y ³ 0);
8.17.	+ 2y.
μ = 7x2	+ 2y.

D : x2 + y2 = 9, x2 + y2 =16,

8.4.x = 0, y = 0 (x ³ 0, y ³ 0);

μ = (2x + 5y)(x2 + y2 ).

D : x2 + y2 =1, x2 + y2 =16,

8.6.x = 0, y = 0 (x ³ 0, y ³ 0);

μ = (x + y)(x2 + y2 ).

D : x2 + y2 = 4, x2 + y2 = 25,

8.8.x = 0, y = 0 (x ³ 0, y £ 0);

μ = (2x - 3y)(x2 + y2 ).

D : x2 + y2 =1, x2 + y2 = 9,

8.10.x = 0, y = 0 (x ³ 0, y £ 0);

μ = (x - y)(x2 + y2 ).

D : x2 + y2 = 9, x2 + y2 = 25,

8.12.x = 0, y = 0 (x £ 0, y ³ 0);

μ = (2y - x)(x2 + y2 ).

D : x2 + y2 = 4, x2 + y2 =16,

8.14.x = 0, y = 0 (x £ 0, y ³ 0);

μ = (2y - 3x)(x2 + y2 ).

D : x2 + y2 = 9, x2 + y2 =16,

8.16.x = 0, y = 0 (x £ 0, y ³ 0);

μ = (2y - 5x)(x2 + y2 ).

D : x2 + y2 =1, x2 + y2 =16,

8.18.x = 0, y = 0 (x ³ 0, y ³ 0);

μ = (x + 3y)(x2 + y2 ).

173

D : x = 2, y2 = 2x, y = 0 ( y ³ 0);

8.19.

μ = 7x2 4 + y2.

D : x = 2, y = 0, y2 = 2x		( y ³ 0);
8.21.	+ y.
μ = 7x2 4	+ y.

D : x = 2, y = 0, y2 = x 2		( y ³ 0);
8.23.	+ 8y.
μ = 7x2 2	+ 8y.

D : x =1, y = 0, y2	= 4x	( y ³ 0);
8.25.
μ = 6x + 3y2 .

D : x = 2, y = 0, y2	= x 2	( y ³ 0);
8.27.
μ = 4x + 6y2.

D : x =	1	, y = 0, y2	= 2x	( y ³ 0);
D : x =	2	, y = 0, y2	= 2x	( y ³ 0);
8.29.	2
μ = 4x + 9y2.
D : x =	1	, y = 0, y2	=16x	( y ³ 0);
8.31.	4

μ =16x + 9y2 2.

D : x2 + y2 =1, x2 + y2 = 4,

8.20.x = 0, y = 0 (x ³ 0, y ³ 0);

μ = (x + 2y)(x2 + y2 ).

D : x2 + y2 =1, x2 + y2 = 9,

8.22.x = 0, y = 0 (x ³ 0, y £ 0);

μ = (2x - y)(x2 + y2 ).

D : x2 + y2 =1, x2 + y2 = 25,

8.24.x = 0, y = 0 (x ³ 0, y £ 0);

μ = (x - 4y)(x2 + y2 ).

D : x2 + y2 = 4, x2 + y2 =16,

8.26.x = 0, y = 0 (x ³ 0, y £ 0);

μ = (3x - y)(x2 + y2 ).

D : x2 + y2 = 4, x2 + y2 = 9,

8.28.x = 0, y = 0 (x £ 0, y ³ 0);

μ = ( y - 4x)(x2 + y2 ).

D : x2 + y2 = 4, x2 + y2 = 9,

8.30.x = 0, y = 0 (x £ 0, y ³ 0);

μ = ( y - 2x)(x2 + y2 ).

Задача 9. Пластинка D задана неравенствами, μ - поверхностная плотность. Найти

массу пластинки.

174

9.1.D : x2 + y2 4 £1;

μ= y2.

D : x2 9 + y2 25 £1;

9.3.y ³ 0;

μ = x2 y.

D : 1£ x2 9 + y2 4 £ 4;

9.5.y ³ 0, y £ x2; μ = 8 yx3 .

9.7.D : x24 + y2 £1;

μ= 4y4.

D : 1£ x2 16 + y2 4 £ 4;

9.9.x ³ 0, y £ x2;

μ = x y.

D : x24 + y2 £1;

9.11.x ³ 0, y ³ 0; μ = 6x3 y3.

9.13.D : x29 + y2 4 £1;

μ= x2 y2.

D : x24 + y2 £1;

9.15.x ³ 0, y ³ 0; μ = 30x3 y7 .

D : 1£ x2 9 + y2 4 £ 2;

9.2.y ³ 0, y £ 23 x;

μ = yx.

D : x2 9 + y2 25 £1;

9.4.y ³ 0;

μ = 7x2 y18.

D : x2 9 + y2 £1;

9.6.x ³ 0;

μ = 7xy6 .

D : 1£ x2 4 + y2 9 £ 4;

9.8.x ³ 0, y £ 3x2;

μ = x y.

D : x24 + y2 9 £1;

9.10.x ³ 0, y ³ 0;

μ = x3 y.

D : 1£ x24 + y2 £ 25;

9.12.x ³ 0, y £ x2;

μ = x y3 .

D : x216 + y2 £1;

9.14.x ³ 0, y ³ 0; μ = 5xy7 .

D : 1£ x29 + y2 4 £ 3;

9.16.y ³ 0, y £ 23 x;

μ = yx.

175

D : x2 + y2 25 £1;

9.17.y ³ 0;

μ = 7x4 y.

9.19.D : x24 + y2 9 £1;

μ= x2.

D : x2 9 + y2 £1;

9.21.x ³ 0;

μ =11xy8.

D : 1£ x2 9 + y2 4 £ 5;

9.23.x ³ 0, y £ 2x3;

μ = x y.

9.25.D : x2 4 + y2 25 £1;

μ= x4.

D : 1£ x2 4 + y2 9 £ 36;

9.27.x ³ 0, y ³ 32 x;

μ = 9 x y3 .

D : x216 + y2 £1;

9.29.x ³ 0, y ³ 0; μ =105x3 y9 .

D : 1£ x2 16 + y2 £ 3;

9.31.x ³ 0, y ³ x4;

μ = x y5 .

D : x2 + y2 9 £1;

9.18.y ³ 0;

μ = 35x4 y3.

D : 1£ x2 + y2 16 £ 9;

9.20.y ³ 0, y £ 4x;

μ = yx3 .

D : 1£ x2 4 + y2 16 £ 5;

9.22.x ³ 0, y £ 2x;

μ = x y.

D : x24 + y2 9 £1;

9.24.x ³ 0, y ³ 0;

μ = x5 y.

D : x2 + y2 4 £1;

9.26.x ³ 0, y ³ 0; μ =15x5 y3.

D : x2100 + y2 £1;

9.28.x ³ 0, y ³ 0; μ = 6xy9 .

D : 1£ x2 9 + y2 16 £ 2;

9.30.y ³ 0, y £ 43 x; μ = 27 yx5 .

176

Задача 10. Найти объем тела, заданного ограничивающими его поверхностями.

10.1. y =16

y =

10.2. y = 5

, y = 5x 3,

z = 0, x + z = 2.

z = 0, z = 5 + 5

x2 + y2

= 2, y =

, y = 0,

x + y = 2, y =

10.3.

10.4.

z = 0, z =15x.

z =12y, z = 0.

x = 5

x = 5y 6,

x = 20 2y, x = 5 2y,

10.5.

10.6. z = 0,

z = 5 (3 +

z = 0,

z + y =1 2.

x2 + y2

= 2, x =

, x = 0,

x + y = 2, x =

10.7.

10.8.

z = 0, z = 30y.

z =12x 5, z = 0.

y = 5

3, y = 5x 9,

y =17

, y = 2

10.9.

z = 0,

x + z =1 2.

10.10. z = 0,

z = 5(3 +

) 9.

x2 + y2 = 8, y =

, y = 0,

x + y = 4, y =

10.11.

10.12.

z = 0,

z =15x 11.

z = 3y, z = 0.

x = 5

x =

x =19 2y, x = 4 2y,

10.13.

10.14.

z = 0,

z =

(3 +

z = 0,

z + y = 2.

x2 + y2 = 8, x =

, x = 0,

x + y = 4, x =

10.15.

10.16.

z = 30y 11, z = 0.

z = 3x 5,

z = 0.

y = 5

y =

y = 6 3x, y = 3x,

10.17.

10.18.

z = 0,

x + z = 3.

z = 0,

z =

(3 +

177

10.19.

x2 + y2

=18,

y =

y = 0,

10.20.

x + y = 6,

y =

z = 0,

z = 5x 11.

= 4y,

z = 0.

x = 5

3, x = 5y 9,

x = 7 3y, x = 2 3y,

10.21.

10.22. z = 0,

z = 5(3 +

) 9.

z = 0,

z + y = 3.

x2 + y2

=18,

x =

x = 0,

x + y = 6,

x =

10.23.

10.24.

z = 0,

z =10y 11.

= 4x 5,

z = 0.

y =

15x

x2 + y2

= 50,

y =

10.25. z = 0,

z =

)

10.26.

y = 0,

z = 0,

z = 3x 11.

(

x + y = 8, y =

10.28. x =16

, x =

10.27.

z = 3y, z = 0.

z + y = 2, z = 0.

x =

, x =15y,

+ y

= 50,

x =

5y,

10.29. z = 0,

z =15(1+

10.30.

x = 0,

z = 0,

z = 6y 11.

10.31. x =17

x = 2

z = 0,

z + y =1 2.

Задача 11. Найти объем тела, заданного ограничивающими его поверхностями.

+ y

= y, x

+ y

= 4y,

+ y

= 2y,

11.1.

11.2.

z = 5 4 - x2 , z = 0.

= x2 + y2 , z = 0.

+ y

= 8

+ y2

+ 4x = 0,

11.3. z = x2 + y2 - 64,

11.4.

= 8 - y2 , z = 0.

z = 0

(z ³ 0).

178

x2 + y2 = 6x, x2 + y2 = 9x,

11.5.z = x2 + y2 , z = 0, y = 0 ( y £ 0)

x2 + y2 = 2y,

11.7. z = 94 - x2 , z = 0.

x2 + y2 + 22y = 0,

11.9.z = x2 + y2 - 4, z = 0 (z ³ 0).

x2 + y2 = 7x, x2 + y2 = 9x,

11.11.z = x2 + y2 , z = 0, y = 0 ( y £ 0)

x2 + y2 = 2y,

11.13.	- x2 , z = 0.
z =13 4

x2 + y2 = 62x,

11.15.z = x2 + y2 - 36, z = 0 (z ³ 0).

x2 + y2 = 4x,

11.17.	- y2 , z = 0.
z =12

x2 + y2 = 42x,

11.19.z = x2 + y2 -16, z = 0 (z ³ 0).

x2 + y2 = 62y,

11.6.z = x2 + y2 - 36, z = 0 (z ³ 0).

x2 + y2 = 2y, x2 + y2 = 5y,

11.8. z =
11.8. z =		x2 + y2 ,	z = 0.
x2	+ y2 = 4x,
11.10.			z = 0.
z =10 - y2 ,			z = 0.

x2 + y2 = 82y,

11.12.z = x2 + y2 - 64, z = 0 (z ³ 0).

x2 + y2 = 3y, x2 + y2 = 6y,

11.14. z = x2 + y2 , z = 0.

x2 + y2 = 22y,

11.16.z = x2 + y2 - 4, z = 0 (z ³ 0).

x2 + y2 = 8x, x2 + y2 =11x,

11.18.z = x2 + y2 , z = 0, y = 0 ( y £ 0)

11.20.x2 + y2 = 4y,

z = 4 - x2 , z = 0.

179

x2 + y2 = 4y, x2 + y2 = 7y,

11.21. z = x2 + y2 , z = 0.

x2	+ y2 + 2x = 0,
11.23.		= 0.
z =17 4 - y2 , z		= 0.

x2 + y2 + 22x = 0,

11.25.z = x2 + y2 - 4, z = 0 (z ³ 0).

x2 + y2 =10x, x2 + y2 =13x,

11.27.z = x2 + y2 , z = 0, y = 0 ( y ³ 0)

x2 + y2 = 2x,

11.29. z = 214 - y2 , z = 0.

x2 + y2 + 2x = 0,

11.31. z = 254 - y2 , z = 0.

x2 + y2 = 42y,

11.22.z = x2 + y2 -16, z = 0 (z ³ 0).

x2 + y2 = 9x, x2 + y2 =12x,

11.24.z = x2 + y2 , z = 0, y = 0 ( y ³ 0)

11.26.x2 + y2 = 4y,

z = 6 - x2 , z = 0.

x2 + y2 = 22x,

11.28.z = x2 + y2 - 4, z = 0 (z ³ 0).

x2 + y2 = 5y, x2 + y2 = 8y,

11.30. z = x2 + y2 , z = 0.

Задача 12. Найти объем тела,

y = 5x2 + 2, y = 7,

12.1.z = 3y2 - 7x2 - 2, z = 3y2 - 7x2 - 5.

x= -5y2 + 2, x = -3,

12.3.z = 3x2 + y2 +1,

z= 3x2 + y2 - 5.

заданного ограничивающими его поверхностями.

y = 5x2 - 2, y = -4x2 + 7,

12.2.z = 4 + 9x2 + 5y2 , z = -1+ 9x2 + 5y2.

x = 2y2 - 3, x = -7 y2 + 6,

12.4.z =1+ x2 +16y2 , z = -3 + x2 +16y2 .

180

y = −6x2 + 8, y = 2,

12.5.z = x − x2 − y2 −1, z = x − x2 − y2 − 5.

x = 5y2 − 9, x = −4,

12.7.z = x2 + 4x − y2 − 4, z = x2 + 4x − y2 + 2.

x= 5y2 −1, x = −3y2 +1,

12.9.z = 2 − x2 + 6y2 ,

z= −1− x2 + 6y2 .

y = −5x2 + 3, y = −2,

12.11.z = 2x2 − 3y − 6y2 −1, z = 2x2 − 3y − 6y2 + 2.

x = 3y2 − 5, x = −2,

12.13.z = 2 − x2 +16y2 , z = 8 − x2 +16y2 .

y = 2x2 −1, y =1,

12.15.z = x2 − 5y2 − 3, z = x2 − 5y2 − 6.

x= −4y2 +1, x = −3,

12.17.z = x2 − 7 y2 −1,

z= x2 − 7 y2 + 2.

y =1− 2x2 , y = −1,

12.19.z = x2 + 2y + y2 − 2, z = x2 + 2y + y2 +1.

y= 5x2 −1, y = −3x2 +1,

12.6.z = −2 + 3x2 + y2 , z = −5 + 3x2 + y2 .

y= 6x2 −1, y = 5,

12.8.z = 2x2 + x − y2 ,

z = 2x2 + x − y2 + 4.

x = −3y2 + 7, x = 4,

12.10.z = 2 + 6x2 + y2 , z = 3 + 6x2 + y2 .

y = x2 − 5, y = −x2 + 3,

12.12.z = 4 + 5x2 + 8y2 , z =1+ 5x2 + 8y2 .

x = y2 − 2, x = −4y2 + 3,

12.14.z = 16 − x2 − y2 + 2, z = 16 − x2 − y2 −1.

y = x2 − 2, y = −4x2 + 3,

12.16.z = 2 + x2 + y2 , z = −1+ x2 + y2 .

x = 7 y2 − 6, x = −2y2 + 3,

12.18.z = 3 −12y2 + 5x2 , z = −2 −12y2 + 5x2.

y = x2 − 7, y = −8x2 + 2,

12.20.z = 3 −12y2 + 5x2 , z = −2 −12y2 + 5x2.

181

x = 2y2 + 3, x = 5,

12.21.z =1+ 9x2 + 4y2 , z = 4 + 9x2 + 4y2 .

x = 5y2 − 2, x = −4y2 + 7,

12.23.z = 4 − 2x2 + 3y2 , z = −1− 2x2 + 3y2 .

y = −3x2 + 5, y = 2,

12.25.z = 3 + 5x2 + y2 , z = −1+ 5x2 + y2 .

x = 4y2 + 2, x = 6,

12.27.z = x2 + 4y2 + y +1, z = x2 + 4y2 + y + 4.

y = 2x2 − 5, y = −3,

12.29.z = 2 + x2 + 4y2 , z = −1+ x2 + 4y2 .

y = −2x2 + 7, y = 5,

12.31.z =1− 2x2 + 3y2 , z = 4 − 2x2 + 3y2.

y = 3x2 + 4, y = 7,

12.22.z = 5 − 2x2 + 3y2 , z =1− 2x2 + 3y2 .

x = −2y2 + 5, x = 3,

12.24.z = 5 − x2 + 25y2 , z = 2 − x2 + 25y2 .

y = 3x2 − 5, y = −6x2 + 4,

12.26.z = 2 +10x2 − y2 , z = −2 +10y2 − y2.

x = 3y2 − 2, x = −4y2 + 5,

12.28.z = 4 − 7x2 − 9y2 , z =1− 7x2 − 9y2.

y = 2x2 − 3, y = −7x2 + 6,

12.30.z =1− 5x2 − 6y2 , z = −3 − 5x2 − 6y2.

Задача 13. Найти объем тела, заданного ограничивающими его поверхностями.


13.1.	z = 9 − x2	− y2	,	13.2.	z =15 x2	+ y2	2,
13.1.	9z 2 = x2 + y2.			13.2.	z =17 2 − x2 − y2.
	9z 2 = x2 + y2.				z =17 2 − x2 − y2.

182

	z =	4 − x2 − y2 ,
	13.3.
	13.3.	(x2 + y2 ) 255.
	z =	(x2 + y2 ) 255.

13.5.z = 169 − x2 − y2 , 2z = x2 + y2.

z =	25 − x2 − y2 ,
13.7.
z =	(x2 + y2 ) 99.

13.9.z = 21x2 + y2 2, z = 232 − x2 − y2.

z =	9 − x2 − y2 ,
13.11.
z =	(x2 + y2 ) 80.

13.13. z = 1− x2 − y2 ,

3z2 = x2 + y2.

	z =	36 − x2 − y2 ,
	13.15.
	13.15.	(x2 + y2 ) 63.
	z =	(x2 + y2 ) 63.

13.17. z = 144 − x2 − y2 ,

18z = x2 + y2.

z =	9 − x2 − y2 ,
13.19.
z =	(x2 + y2 ) 35.

z = 64 − x2 − y2 , z =1,

13.4. x2 + y2 = 60

(внутри цилиндра).

13.6.z = 3x2 + y2 , z =10 − x2 − y2.

z = 100 − x2 − y2 , z = 6,

13.8. x2 + y2 = 51

(внутри цилиндра).

13.10.z = 16 − x2 − y2 , 6z = x2 + y2.

z = 81− x2 − y2 , z = 5,

13.12. x2 + y2 = 45

(внутри цилиндра).

13.14.z = 6x2 + y2 , z =16 − x2 − y2.

z = 64 − x2 − y2 , z = 4,

13.16. x2 + y2 = 39

(внутри цилиндра).

13.18.z = 3x2 + y2 2, z = 52 − x2 − y2.

z = 49 − x2 − y2 , z = 3,

13.20. x2 + y2 = 33

(внутри цилиндра).

183

13.21. z = 36 - x2 - y2 ,

9z = x2 + y2.

z =	16		- x2 - y2 ,
13.23.
		(x	2 + y2 ) 15.
z =

13.25.z = 49 - x2 - y2 , z = x2 + y2.

	z =	9 - x2 - y2 ,
	13.27.
	13.27.	(x2 + y2 ) 8.
	z =	(x2 + y2 ) 8.

13.29. z = 64 - x2 - y2 ,

12z = x2 + y2.

	z =	36 - x2 - y2 ,
	13.31.
	13.31.	(x2 + y2 ) 3.
	z =	(x2 + y2 ) 3.

13.22.z = 9x2 + y2 , z = 22 - x2 - y2.

z = 36 - x2 - y2 , z = 2,

13.24. x2 + y2 = 27

(внутри цилиндра).

13.26.z =12x2 + y2 , z = 28 - x2 - y2.

z = 25 - x2 - y2 , z =1,

13.28. x2 + y2 = 21

(внутри цилиндра).

13.30.z = 9x2 + y2 2, z =112 - x2 - y2.

Задача 14. Найти объем тела,

14.1.z = 2 -12(x2 + y2 ), z = 24x + 2.

14.3.z = 8(x2 + y2 ) + 3, z =16x + 3.

14.5.z = 4 -14(x2 + y2 ), z = 4 - 28x.

заданного ограничивающими его поверхностями.

14.2.z =10 éë(x -1)2 + y2 ùû +1, z = 21- 20x.

14.4.z = 2 - 20éë(x +1)2 + y2 ùû, z = -40 - 38x.

14.6.z = 28éë(x +1)2 + y2 ùû + 3, z = 56x + 59.

184

14.7.z = 32(x2 + y2 ) + 3, z = 3 - 64x.

14.9.z = 2 - 4(x2 + y2 ), z = 8x + 2.

14.11.z = 24(x2 + y2 ) +1, z = 48x +1.

14.13.z = -16(x2 + y2 ) -1, z = -32x -1.

14.15.z = 26(x2 + y2 ) - 2, z = -52x - 2.

14.17.z = -2(x2 + y2 ) -1, z = 4y -1.

14.19.z = 30(x2 + y2 ) +1, z = 60y +1.

14.21.z = 2 -18(x2 + y2 ), z = 2 - 36y.

14.23.z = 22(x2 + y2 ) + 3, z = 3 - 44y.

14.25.z = 4 - 6(x2 + y2 ), z =12y + 4.

14.27.z = 28(x2 + y2 ) + 3, z = 56y + 3.

14.8.z = 4 - 6éë(x -1)2 + y2 ùû, z =12x -8.

14.10.z = 22éë(x -1)2 + y2 ùû + 3, z = 47 - 44x.

14.12.z = 2 -18éë(x +1)2 + y2 ùû, z = -36x - 34.

14.14.z = 30éë(x +1)2 + y2 ùû +1, z = 60x + 61.

14.16.z = -2éë(x -1)2 + y2 ùû -1, z = 4x - 5.

14.18.z = 26éë(x -1)2 + y2 ùû - 2, z = 50 - 52x.

14.20.z = -16éë(x +1)2 + y2 ùû -1, z = -32x - 33.

14.22.z = 24éë(x +1)2 + y2 ùû +1, z = 48x + 49.

14.24.z = 2 - 4éë(x -1)2 + y2 ùû, z = 8x - 6.

14.26.z = 32éë(x -1)2 + y2 ùû + 3, z = 67 - 64x.

14.28.z = 4 -14éë(x +1)2 + y2 ùû, z = -28x - 24.

185

z = 2 - 20(x

+ y

z = 8

+ y

2 ù

+ 3,

14.29.

14.30.

ë(x +1)

z = 2 - 40y.

z =16x +19.

14.31.z =10(x2 + y2 ) +1, z =1- 20y.

Задача 15. Найти объем тела, заданного неравенствами.

1 £ x2 + y2 + z2							£ 49,

15.1. -	x2	+ y2			£ z £					x2				+ y2	,
15.1. -		35			£ z £									3	,
		35												3
-x £ y £ 0.
4 £ x2 + y2 + z2 £ 64,
15.3. z £
		x2 + y2				,	-			x				£ y £ 0.

			3							3
			3							3
1 £ x2 + y2 + z2							£ 36,
15.5. z ³
		x2 + y2				,	-						x £ y £				x.
		x2 + y2									3					3
											3					3
		99
1 £ x2 + y2 + z2 £ 49,

15.7. 0 £ z £				x2	+ y2
								,
					24			,
					24
y £ -		x		, y £ -								x.
		x							3

		3
		3
4 £ x2 + y2 + z2 £ 64,

15.9. -	x2	+ y2			£ z £					x2				+ y2	,
15.9. -		35			£ z £									3	,
		35												3

x £ y £ 0.

	1 £ x2 + y2 + z2 £ 64,

15.2.			x2			+ y2				£ z £			x2 + y					2	,
15.2.					15					£ z £			3						,
					15								3
	-					x £ y £ 0.
	-			3		x £ y £ 0.
	4 £ x2 + y2 + z2 £ 36,
15.4. z ³ -
								x2 + y2				,	0 £ y £ -								x	.
																						.
									63												3
									63												3
	25 £ x2 + y2 + z2 £100,
15.6. z £ -
								x2 + y2				,					x £ y £ -							x.
								x2 + y2							3								3
															3								3
									99
	25 £ x2 + y2 + z2 £ 49,

				x2			+ y2
15.8. -											£ z £ 0,
15.8. -							24				£ z £ 0,
							24
	y ³ -							x	, y ³ -							x.
								x						3

							3
							3
		16 £ x2 + y2 + z2 £100,

				x2			+ y		2		£ z £			x2 + y2
15.10.																				,
15.10.					15								3							,
					15								3

3x £ y £ 0.

186

16 £ x2 + y2 + z2 £			100,					16 £ x2 + y2 + z2 £ 64,

								15.12. z ³ -			x2 + y2	,
15.11.
	x2 + y2				x
	x2 + y2				x					63
z £		, -		3x £ y £ -
	3						.
						3	.		x
						3		-		£ y £ -			3x.


								3
								3

4	£ x2 + y2 + z2 £ 49,
15.13. z ³
			x2 + y2					,	y £ 0,					y £			x.
			x2 + y2													3
																3
					99
4	£ x2 + y2 + z2 £ 64,

	£ z £					x2	+ y2
15.15. 0										,
15.15. 0							24			,
							24
y £				x, y £					x		.
			3						x

									3
									3
9	£ x2 + y2 + z2 £ 81,

15.17. -		x2 + y2					£ z £					x2		+ y2	,
15.17. -		3					£ z £						35		,
		3											35
0	£ y £ -x.

36 £ x2 + y2 + z2			£144,
15.19. z £
	x2 + y2	,			x £ y £	x	.
	x2 + y2			3		x
	3
						3
						3

36 £ x2 + y2 + z2 £121,

15.14. z ³ -			x2 + y2				,		y ³ 0,			y ³		x.
			x2 + y2										3
			99										3
			99
36 £ x2 + y2 + z2 £144,

15.16. -	x2	+ y2			£ z £ 0,
15.16. -		24			£ z £ 0,
		24
y ³			x, y ³				x				.
		3					x
		3
								3
		36 £ x2 + y2 + z2 £144,

15.18. -				x2	+ y2				£ z £			x2 + y2			,
15.18. -					3				£ z £			35			,
					3							35
		0 £ y £ -								x.
		0 £ y £ -						3		x.
36 £ x2 + y2 + z2 £100,

15.20. z ³ -			x2 + y2			,
15.20. z ³ -			63			,
			63

x3 £ y £ 3x.

9 £ x2 + y2 + z2 £ 64,								49 £ x2 + y2 + z2 £144,

15.21. z ³		x2		+ y2	,			15.22. z £ -		x2		+ y2	,
15.21. z ³	99				,			15.22. z £ -				99	,
	99											99
y £	x		,	y £ -		x	.	y ³	x		,	y ³ -		x	.

	3					3			3					3
	3					3			3					3

187

9 £ x2 + y2 + z2 £ 81,															49 £ x2 + y2 + z2 £ 81,

15.23. 0 £ z £				x2	+ y2					,					15.24. -	x2	+ y2			£ z £ 0,
15.23. 0 £ z £					24					,					15.24. -		24			£ z £ 0,
					24												24
y £ 0, y £						x		.							y ³ 0, y ³						x				.

						3														3
						3														3
16 £ x2 + y2 + z2 £100,																	64 £ x2 + y2 + z2 £196,

15.25. -	x2	+ y2			£ z £							x2 + y2		,	15.26. -				x2		+ y2							£ z £ -	x2 + y2		,
15.25. -		3			£ z £					35				,	15.26. -					3								£ z £ -	15		,
		3								35										3									15
0 £ y £ x.																	0 £ y £										x.
0 £ y £ x.																	0 £ y £							3			x.
64 £ x2 + y2 + z2										£196,					64 £ x2 + y2 + z2													£144,
15.27. z £															15.28. z ³ -
		x2 + y2				,					x		£ y £ 0.					x2 + y2				,						0 £ y £	x	.
			3															63												.
										3																			3
										3																			3
16 £ x2 + y2 + z2 £ 81,															64 £ x2 + y2 + z2 £169,

15.29. z ³		x2	+ y2			,									15.30. z £ -			x2 + y2				,
15.29. z ³			99			,									15.30. z £ -			99				,
			99															99
y £ 0, y £ -									x.						y ³ 0, y ³ -											x.
y £ 0, y £ -							3		x.						y ³ 0, y ³ -								3			x.
16 £ x2 + y2 + z2 £100,

15.31. 0 £ z £				x2	+ y2					,
15.31. 0 £ z £					24					,
					24
y £ 0,			y £			x		.

						3
						3

188

Задача 16. Тело V задано ограничивающими его поверхностями, μ - плотность.

Найти массу тела.

64(x2 + y2 ) = z2 , x2 + y2 = 4,

16.1. y = 0,		z = 0	( y ³ 0, z ³ 0),
μ = 5(x2 + y2 )				4.
x2 + y2 + z2 = 4, x2 + y2 =1,
16.2. (x2 + y2 £1),			x = 0 (x ³ 0);
μ = 4	z	.

x2 + y2 =1, x2 + y2 = 2z,
16.3. x = 0,	y = 0,		z = 0		(x ³ 0,	y ³ 0);
μ =10x.
x2 + y2 = 16 z2 ,				x	2 + y2 =	4 z,
	49					7
16.4. x = 0,	y = 0,		(x ³ 0, y ³ 0);
μ = 80yz.
x2 + y2 + z2 =1, x2 + y2 = 4z2 ,
16.5. x = 0,	y = 0,		(x ³ 0, y ³ 0, z ³ 0);
μ = 20z.
36(x2 + y2 ) = z2 , x2 + y2 =1,
16.6. x = 0,		z = 0	(x ³ 0, z ³ 0),
μ = 5	(x2 + y2 ).
6
x2 + y2 + z2 =16,					x2 + y2 = 4,

16.7.(x2 + y2 £ 4);

μ= 2 z .

189

x2 + y2 = 4, x2 + y2 = 8z,

16.8. x = 0, y = 0,

z = 0

(x ³ 0,

y ³ 0);

μ = 5x.

x2 + y2 =

, x

2 + y2 = 2 z,

16.9. x = 0,

y = 0,

(x ³ 0, y ³ 0);

μ = 28xz.

x2 + y2 + z2 = 4, x2 + y2 = z2 ,

16.10. x = 0,

y = 0,

(x ³ 0, y ³ 0, z ³ 0);

μ = 6z.

25(x2 + y2 ) = z2 , x2 + y2 = 4,

x = 0,

y = 0,

z = 0

16.11. (x ³ 0, y ³ 0,

z ³ 0),

μ = 2(x2 + y2 ).

x2 + y2 + z2 = 9, x2 + y2 = 4,

16.12. (x2 + y2 £ 4),

y = 0 ( y ³ 0);

μ =

x2 + y2 =1, x2 + y2 = 6z,

16.13. x = 0,

y = 0,

z = 0 (x ³ 0,

y ³ 0);

μ = 90y.

x2 + y

2 =

z2 , x2 + y2 =

1 z,

16.14. x = 0,

y = 0,

(x ³ 0, y ³ 0);

μ =14yz.

190

x2 + y2 + z2 = 4, x2 + y2 = 9z2 ,

16.15. x = 0,			y = 0,			(x ³ 0,	y ³ 0,	z ³ 0);
μ =10z.
9(x2 + y2 ) = z2 , x2 + y2 = 4,
x = 0,			y = 0,			z = 0
16.16. (x ³ 0,			y ³ 0,			z ³ 0),
μ = 5(x2 + y2 ) 3.
x2 + y2 + z2 = 4,
16.17. x2 + y2			=1,		(x2 + y2 £1);
μ =	z	.

x2 + y2 =1, x2 + y2 = z,
x = 0,			y = 0,			z = 0,
16.18. (x ³ 0,			y ³ 0);
μ =10y.
x2 + y2 =				1	z2 , x2 + y2 = 1 z,

			49				7
16.19. x = 0,			y = 0,			(x ³ 0,	y ³ 0);
μ =10xz.
x2 + y2 + z2 = 4, x2 + y2 = 4z2 ,
16.20. x = 0,			y = 0,			(x ³ 0,	y ³ 0,	z ³ 0);
μ =10z.
16(x2 + y2 ) = z2 , x2 + y2 =1,
16.21. x = 0,			y = 0,			z = 0	(x ³ 0,	y ³ 0, z ³ 0),

μ = 5(x2 + y2 ).

191

x2 + y2 + z2				=16,
16.22. x2 + y2			= 4	(x2 + y2 £ 4);
μ =	z	.

x2 + y2 = 4, x2 + y2 = 4z,
16.23. x = 0,			y = 0,		z = 0	(x ³ 0,	y ³ 0);
μ = 5y.
x2 + y2 = z2 ,					x2 + y2 = z,
16.24. x = 0,			y = 0,		(x ³ 0,	y ³ 0);
μ = 35yz.
x2 + y2 + z2 =1, x2 + y2 = z2 ,
16.25. x = 0,			y = 0,		(x ³ 0,	y ³ 0,	z ³ 0);
μ = 32z.
x2 + y2 = z2 , x2 + y2 = 4,
x = 0,			y = 0,		z = 0
16.26. (x ³ 0,			y ³ 0,		z ³ 0),
μ = 5(x2 + y2 ) 2.
x2 + y2 + z2 = 9, x2 + y2 = 4,
16.27. (x2 + y			2 £ 4),		z = 0	(z ³ 0);
μ = 2z.
x2 + y2 =1, x2 + y2 = 3z,
x = 0,			y = 0,		z = 0
16.28. (x ³ 0,			y ³ 0);

μ =15x.

192

x2 + y2 =		4	z2 , x2 + y2 = 2 z,

	49			7
16.29. x = 0,	y = 0,		(x ³ 0,	y ³ 0);
μ = 20xz.
x2 + y2 + z2 =16, x2 + y2 = 9z2 ,
16.30. x = 0,	y = 0,		(x ³ 0,	y ³ 0, z ³ 0);
μ = 5z.
4(x2 + y2 ) = z2 , x2 + y2 =1,
16.31. y = 0,	z = 0		( y ³ 0,	z ³ 0),
μ =10(x2 + y2 ).

193

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