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Воронежский государственный технический университет

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Высшая математика

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Высшая математика. Кратные интегралы и векторный анализ. практикум. Пантелеев И.Н

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4. ЗАДАЧИ ДЛЯ САМОСТОЯТЕЛЬНОГО РЕШЕНИЯ 4.1. Задачи по разделу кратные интегралы

Задача 1. Вычислить.

1.1.	∫∫(12x2 y2 +16x3 y3 )dxdy;	1.2.	∫∫(9x2 y2 +48x3 y3 )dxdy;
	D		D

	D : x =1, y = x2 , y = − x.
1.3.	∫∫(36x2 y2 −96x3 y3 )dxdy;	1.4.
1.3.	D	1.4.
	D : x =1, y = 3 x, y = −x3.
1.5.	∫∫(27x2 y2 + 48x3 y3 )dxdy;	1.6.
1.5.	D	1.6.
	D : x =1, y = x2 , y = −3 x.
1.7.	∫∫(18x2 y2 +32x3 y3 )dxdy;	1.8.
1.7.	D	1.8.

D : x =1, y = x, y = −x2 .

∫∫(18x2 y2 +32x3 y3 )dxdy;

D : x =1, y = x3 , y = −3 x.

∫∫(18x2 y2 +32x3 y3 )dxdy;

D : x =1, y = 3 x, y = −x2 .

∫∫(27x2 y2 + 48x3 y3 )dxdy;

	D : x =1, y = x3 , y = − x.	D : x =1, y = x, y = −x3.
1.9.	∫∫(4xy +3x2 y2 )dxdy;	1.10.	∫∫(12xy +9x2 y2 )dxdy;
1.9.	D	1.10.	D
	D : x =1, y = x2 , y = − x.		D : x =1, y = x, y = −x2 .
	∫∫(8xy +9x2 y2 )dxdy;	1.12.	∫∫(24xy +18x2 y2 )dxdy;
1.11. D		1.12.	D

D : x =1, y = 3 x,

∫∫(12xy +27x2 y2

1.13.D

D :	x =1,		y = x2 ,
	4	xy +	9	x	2	y	2


1.15. ∫∫D	5		11

y = −x3. )dxdy;

y = −3 x.

dxdy;

D : x =1, y = x3 , y = −3 x.

∫∫(8xy +18x2 y2 )dxdy;

1.14.D

D : x =1, y = 3 x, y = −x2 .

1.16.	∫∫D	4	2	2
1.16.	∫∫D	5	xy +9x y
			xy +9x y	dxdy;

D : x =1, y = x3 , y = − x.

D : x =1, y = x, y = −x3.

191

1.17.	∫∫(	24xy −48x3 y3 )dxdy;		1.18.
	D
	D : x =1, y = x2 , y = − x.
1.19.	∫∫(	4xy +16x3 y3 )dxdy;		1.20.
	D
	D : x =1, y = 3 x, y = −x3.
1.21.	∫∫(	44xy +16x3 y3 )dxdy;		1.22.
	D
	D : x =1, y = x2 , y = −3 x.
1.23.	∫∫(xy −4x3 y3 )dxdy;			1.24.
	D
	D : x =1, y = x3 , y = − x.
		6x2 y2 + 25 x4 y4	dxdy;
1.25. ∫∫D		3		1.26.
	D : x =1, y = x2 , y = − x.
		3x2 y2 + 50 x4 y4	dxdy;
1.27. ∫∫D		3		1.28.
	D : x =1, y = 3 x, y = −x3.
1.29.	∫∫(54x2 y2 +150x4 y4 )dxdy;
	D			1.30.

D : x =1, y = x2 , y = −3 x.

∫∫(54x2 y2 +150x4 y4 )dxdy;

1.31.D

D : x =1, y = x3 , y = − x.

∫∫(6xy + 24x3 y3 )dxdy;

D : x =1, y = x, y = −x2 .

∫∫(4xy +16x3 y3 )dxdy;

D : x =1, y = x3 , y = −3 x.

∫∫(4xy +176x3 y3 )dxdy;

D : x =1, y = 3 x, y = −x3.

∫∫(4xy +176x3 y3 )dxdy;

D : x =1, y = x, y = −x3.

∫∫(9x2 y2 + 25x4 y4 )dxdy;

D : x =1, y = x, y = −x2 .

∫∫(9x2 y2 + 25x4 y4 )dxdy;

D : x =1, y = x3 , y = −3 x.

∫∫(xy −9x5 y5 )dxdy;

D : x =1, y = 3 x, y = −x2 .

192

Задача 2. Вычислить.

∫∫yexy/2dxdy;

∫∫y

dxdy;

2.1.

sin 2

2.2. D

D : y = ln 2, y = ln 3, x =

2, x = 4.

D : x = 0, y = π , y =

2.3.

∫∫y cos xy dxdy;

2.4.

∫∫y2e−xy/ 4dxdy;

D : y =π / 2, y =π, x =1, x = 2.

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

2.5.

∫∫y sin xy dxdy;

2.6.

∫∫y2 cos xy

dxdy;

D : y =π / 2, y =π, x =1, x = 2.

D : x = 0, y = π 2, y = x 2.

∫∫4 ye2 xy dxdy;

∫∫4 y2 sin xy dxdy;

2.7. D

2.8. D

π , y = x.

D : y = ln 3, y = ln 4, x =

1 , x =1.

D : x = 0, y =

∫∫y cos 2xy dxdy;

∫∫y2e−xy/8dxdy;

2.9.

2.10.

D : y = π , y =π, x = 1

, x =1.

D : x = 0, y = 2,

y =

∫∫12 y sin 2xy dxdy;

∫∫y

cos xy dxdy;

2.11.

D : y = π , y =

π , x =

2.12. D

2, x = 3.

D : x = 0, y = π , y = x.

∫∫y2 sin 2xy dxdy;

2.13.

∫∫yexy/ 4 dxdy;

2.14.

D : y = ln 2, y = ln 3, x = 4, x =8.

D : x = 0, y = 2π , y = 2x.

∫∫2 y cos 2xy dxdy;

−xy/ 2

dxdy;

2.15.

∫∫y

π , x =1, x = 2.

2.16. D

D : y = π , y =

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

193

∫∫y sin xy dxdy;

∫∫y2 cos 2xy dxdy;

2.17. D

2.18. D

D : y =π, y = 2π, x =

, x =1.

D : x = 0, y =

y =

∫∫8ye4 xy dxdy;

∫∫

3y2 sin

dxdy;

3.19. D

2.20. D

4π

D : y = ln 3, y = ln 4, x =

, x =

: x = 0, y =

y =

∫∫y cos xy dxdy;

∫∫y2e−xy/2dxdy;

2.21.

2.22.

D : y =π, y = 3π, x =1 2, x =1.

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

∫∫ysin 2xy dxdy;

2.23.

∫∫y2 cos xy dxdy;

2.24. D

D : y =π 2, y = 3π 2, x =1 2, x = 2.

D : x = 0, y = π , y = 2x.

∫∫6 yexy/3dxdy;

∫∫y2 sin

dxdy;

2.25.

2.26.

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

D : x = 0, y = π , y = x.

2.27.

∫∫y cos 2xy dxdy;

∫∫y2e−xy/8dxdy;

2.28. D

D : y =π 2, y = 3π 2, x =1 2, x = 2.

D : x = 0, y = 4, y = 2x.

∫∫3ysin xy dxdy;

2.30. ∫∫D

y2 cos

dxdy;

2.29.

D : y =π 2, y = 3π, x =1, x = 3.

D : x = 0, y = 2π , y = 2x.

∫∫12 ye6 xy dxdy;

2.31.D

D : y = ln 3, y = ln 4, x =1 6, x =1 3.

194

Задача 4. Вычислить.

	∫∫∫		2 y2exy		dx dy dz;
3.1.	V	x = 0,			y =1, y = x,
	V			= 0, z		=1.
		z
	∫∫∫y2ch (2xy) dx dy dz;
3.3.	V	x = 0, y = −2, y = 4x,
	V			= 0, z		= 2.
		z
	∫∫∫x2sh (3xy) dx dy dz;
3.5.	V	x =1, y = 2x, y = 0,
	V			= 0, z		= 36.
		z
			2	π
	∫∫∫y			cos	4	xy dx dy dz;
3.7.	V
		x = 0, y = −1, y = x 2,
	V			= 0,	z	= −π2 .
		z
	∫∫∫y2e−xy dx dy dz;
3.9.	V	x = 0, y = −2, y = 4x,
	V			= 0, z		=1.
		z
	∫∫∫y2ch (2xy) dx dy dz;
3.11.		V	x = 0, y =1, y = x,
	V
			z = 0, z = 8.

	∫∫∫x2 z sin (xyz) dx dy dz;
3.2.	V	x = 2, y =π, z =1,
	V			= 0, y =1, z = 0.
		x
	∫∫∫		8 y2 z e2 xyz dx dy dz;
3.4.	V	x		= −1, y = 2, z =1,
	V			= 0, y = 0, z = 0.
		x
	∫∫∫y2 z cos (xyz) dx dy dz;
3.6.	V	x =1, y = 2π, z = 2,
	V			= 0, y =1, z = 0.
		x
	∫∫∫x2 z sin xyz dx dy dz;
3.8.	V			4
		x =1, y = 2π, z = 4,
	V
				= 0, y = 0, z = 0.
		x
	∫∫∫2 y2 z e2 xyz dx dy dz;
3.10.		V	x =1, y =1, z =1,
	V
			x = 0, y = 0, z = 0.
	∫∫∫x2 z sh (xyz) dx dy dz;
3.12.	V	x = 2, y =1, z =1,
	V			= 0, y = 0, z = 0.
		x

195

	∫∫∫y2exy 2			dx dy dz;
3.13.	V	x = 0, y = 2, y = 2x,
	V		= 0,	z = −1.
		z	= 0,	z = −1.
		2		πxy
	∫∫∫y		cos	dx dy dz;
3.15.	V			2
3.15.	V	x = 0,		y = −1, y = x,
	V		= 0,	z = 2π2 .
		z	= 0,	z = 2π2 .
	∫∫∫y2cos (πxy) dx dy dz;
3.17.	V	x = 0,		y =1, y = 2x,
	V		= 0,	z =π2 .
		z	= 0,	z =π2 .

	∫∫∫y2 z cos xyz dx dy dz;
3.14.	V	3
3.14.	V	x = 3, y =1, z = 2π,
	V	x = 3, y =1, z = 2π,
	V
		x = 0, y = 0, z = 0.
	∫∫∫x2 z sh (xyz) dx dy dz;
3.16.	V	x =1, y = −1, z =1,
V
		x = 0, y = 0, z = 0.
∫∫∫2x2 z sh (2xyz) dx dy dz;
3.18. V	x = 2, y =1 2, z =1 2,
V
	x = 0, y = 0, z = 0.

∫∫∫x2sh (2xy) dx dy dz;

3.19. V x = −1, y = x, y = 0,

V z = 0, z = 8.

∫∫∫y2ch (xy) dx dy dz;

3.21. V x = 0, y = −1, y = x,

V z = 0, z = 2.

∫∫∫x2cos π2 xy dx dy dz;

3.23. V x = 2, y = x, y = 0,

V z = 0, z =π.

	∫∫∫x2 z sin xyz dx dy dz;
3.20.	V		2
3.20.	V	x	=1, y = 4, z =π,
	V	x	=1, y = 4, z =π,
	V		= 0, y = 0, z = 0.
		x	= 0, y = 0, z = 0.
∫∫∫x2 z ch (xyz) dx dy dz;
3.22.	V	x =1, y =1, z =1,
V
		x = 0, y = 0, z = 0.
	∫∫∫y		2 z cos xyz dx dy dz;
3.24.	V		9
3.24.	V
	V	x = 9, y =1, z = 2π,
	V
		x = 0, y = 0, z = 0.

196

∫∫∫x2cos (πxy) dx dy dz;

3.25.

x =1, y = 2x, y = 0,

= 0, z = 4π.

∫∫∫V

dx dy dz

;

z 4

3.27. V

=1,

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

∫∫∫x2sin (4πxy)

dx dy dz;

3.29.

x =1, y = x 2, y = 0,

= 0, z = 8π.

∫∫∫x2sh (xy) dx dy dz;

3.31.

x = 2, y = x 2, y = 0,

= 0, z =1.

	∫∫∫y			2		xyz
					z ch		dx dy dz;
3.26.	V					2
		x = 2, y = −1, z = 2,
V
					= 0, y = 0, z = 0.
		x
∫∫∫		2 y2 z ch (2xyz) dx dy dz;
V
3.28.			=		1	, y = 2, z = −1,
V	x				2

			= 0, y = 0, z = 0.
	x
		∫∫∫8 y2 z e−xyz dx dy dz;
3.30.			V		x = 2, y = −1, z = 2,
		V					= 0, z = 0.
					x = 0, y

Задача 4. Найти площадь фигуры, ограниченной данными линиями.

4.1. y = 3

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

4.2. x =

36 − y2 , x = 6 −

36 − y2 .

4.3. x2 + y2 = 72, 6 y = −x2 (y ≤ 0). 4.4. x =8 − y2 , x = −2 y.

4.5. y =

3 , y =8ex , y = 3,

y = 8.

4.6. y =

, y =

, x =16.

(y ≤ 0).

4.7. x = 5 − y2 , x = −4 y.

4.8. x2 + y2

=12,

6 y = x2

4.9. y =

12 − x2 , y = 2 3 −

12 − x2 ,

4.10. y =

x, y =

, x = 9.

x = 0 (x ≥ 0).

197

4.11.	y = 24 − x2 , 2 3y = x2		,
	x = 0	(x ≥ 0).

4.13. y = 20 − x2 , y = −8x. 4.15. y = 32 − x2 , y = −4x.

y = sin x, y = cos x, 4.12. x = 0, (x ≥ 0).

4.14.	y = 18 − x2 , y = 3 2 − 18 − x2 .
4.16.	y = 2 x, y = 5ex , y = 2, y = 5.

x2 + y2

= 36,

2 y = x2

4.18. y = 3

x, y = 3 x, x = 4.

4.17. (y ≥ 0).

4.19. y = 6 −

36 − x2 , y = 36 − x2 ,

4.20. y = 25 4 − x2 , y = x −5 2.

x = 0 (x ≥ 0).

4.21. y =

x, y =1 x, x =16.

4.22. y = 2

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

4.23. x = 27 − y2 , x = −6 y.

4.24.

x = 72 − y2 , 6x = y2 ,

y = 0

(y ≥ 0).

4.25. y =

6 − x2 , y =

6 − 6 − x2 . 4.26.

y = 3

x, y =

, x = 4.

y = sin x, y = cos x,

4.27. x = 0,

(x ≤ 0).

4.28. y = x , y

= 6e

, y =1, y = 6.

4.29. y = 3

x, y = 3 x, x = 9.

4.30. y =11− x2 , y = −10x.

4.31. x2 + y2

=12,

6 = y2

(x ≥ 0).

Задача 5. Пластинка D задана ограничивающими ее кривыми, μ - поверхностная плотность. Найти массу

пластинки.

5.1.D : x =1, y = 0, y2 = 4x

μ= 7x2 + y.

5.3.D : x =1, y = 0, y2 = 4x

μ= 7x2 2 +5y.

(y ≥ 0); 5.2.	D : x2 + y2 =1, x2 + y2 = 4,
(y ≥ 0); 5.2.	x = 0, y = 0 (x ≥ 0, y ≥ 0);
	μ = (x + y) (x2 + y2 ).
(y ≥ 0); 5.4.	D : x2 + y2 = 9, x2 + y2 =16,
(y ≥ 0); 5.4.	x = 0, y = 0 (x ≥ 0, y ≥ 0);

μ = (2x +5y)(x2 + y2 ).

198

	D : x = 2, y = 0, y2		= 2x (y ≥ 0);	D : x2 + y2 =1, x2 + y2 =16,
5.5.	D : x = 2, y = 0, y2		= 2x (y ≥ 0);	5.6.	x = 0, y = 0 (x ≥ 0, y ≥ 0);
	μ = 7x2	8 + 2 y.			μ = (x + y) (x2 + y2 ).
					μ = (x + y) (x2 + y2 ).
	D : x = 2, y = 0, y2		= x 2 (y ≥ 0); 5.8.		D : x2 + y2 = 4, x2 + y2 = 25,
5.7.	D : x = 2, y = 0, y2		= x 2 (y ≥ 0); 5.8.		x = 0, y = 0 (x ≥ 0, y ≤ 0);
	μ = 7x2	2 + 6 y.			μ = (2x −3y) (x2 + y2 ).
					μ = (2x −3y) (x2 + y2 ).

5.9.D : x =1, y = 0, y2 = 4x

μ= x +3y2 .

5.11.D : x =1, y = 0, y2 = x

μ= 3x + 6 y2 .

5.13.D : x = 2, y = 0, y2 = x

μ= 2x +3y2 .

(y ≥ 0);	D : x2 + y2 =1, x2 + y2 = 9,
(y ≥ 0);	5.10.	x = 0, y = 0 (x ≥ 0, y ≤ 0);
		μ = (x − y) (x2 + y2 ).
(y ≥ 0);	D : x2 + y2 = 9, x2 + y2 = 25,
(y ≥ 0);	5.12.	x = 0, y = 0 (x ≤ 0, y ≥ 0);
		μ = (2 y − x) (x2 + y2 ).
2 (y ≥ 0); 5.14.		D : x2 + y2 = 4, x2 + y2 =16,
2 (y ≥ 0); 5.14.		x = 0, y = 0(x ≤ 0, y ≥ 0);

μ = (2 y −3x)(x2 + y2 ).

	D : x =	1	, y	= 0, y2 =8x (y ≥ 0);			D : x2 + y2 = 9, x2 + y2 =16,
5.15.		1				5.16.	x = 0, y = 0 (x ≤ 0, y ≥ 0);
		2
		2
	μ = 7x +3y2 .						μ = (2 y −5x) (x2 + y2 ).
5.17. D : x =1, y = 0, y2					= 4x (y ≥ 0); 5.18.		D : x2 + y2 =1, x2 + y2 =16,
5.17. D : x =1, y = 0, y2					= 4x (y ≥ 0); 5.18.		x = 0, y = 0 (x ≥ 0, y ≥ 0);
	μ = 7x2 + 2 y.						μ = (x +3y) (x2 + y2 ).
							μ = (x +3y) (x2 + y2 ).
5.19. D : x = 2, y2				= 2x, y = 0 (y ≥ 0); 5.20.			D : x2 + y2 =1, x2 + y2 = 4,
5.19. D : x = 2, y2				= 2x, y = 0 (y ≥ 0); 5.20.			x = 0, y = 0 (x ≥ 0, y ≥ 0);
	μ = 7x2			4 + y	2.		μ = (x + 2 y) (x2 + y2 ).
							μ = (x + 2 y) (x2 + y2 ).

199

5.21. D : x = 2, y = 0, y2				= 2x (y ≥ 0);		D : x2 + y2 =1, x2 + y2 = 9,
5.21. D : x = 2, y = 0, y2				= 2x (y ≥ 0);	5.22.		x = 0, y = 0 (x ≥ 0, y ≤ 0);
	μ = 7x2 4 + y.						μ = (2x − y) (x2 + y2 ).
							μ = (2x − y) (x2 + y2 ).
5.23. D : x = 2, y = 0, y2				= x 2 (y ≥ 0); 5.24.			D : x2 + y2 =1, x2 + y2 = 25,
5.23. D : x = 2, y = 0, y2				= x 2 (y ≥ 0); 5.24.			x = 0, y = 0(x ≥ 0, y ≤ 0);
	μ = 7x2 2 +8 y.						μ = (x − 4 y) (x2 + y2 ).
							μ = (x − 4 y) (x2 + y2 ).
5.25. D : x =1, y = 0, y2						D : x2 + y2 = 4, x2 + y2 =16,
5.25. D : x =1, y = 0, y2				= 4x(y ≥ 0); 5.26.			x = 0, y = 0 (x ≥ 0, y ≤ 0);
	μ = 6x +3y2 .						μ = (3x − y) (x2 + y2 ).
							μ = (3x − y) (x2 + y2 ).
5.27. D : x = 2, y = 0, y2				= x 2(y ≥ 0);		D : x2 + y2 = 4, x2 + y2 = 9,
5.27. D : x = 2, y = 0, y2				= x 2(y ≥ 0);	5.28.		x = 0, y = 0 (x ≤ 0, y ≥ 0);
	μ = 4x + 6 y2 .						μ = (y − 4x) (x2 + y2 ).
							μ = (y − 4x) (x2 + y2 ).
	D : x =	1	, y = 0, y2 = 2x (y ≥ 0);				D : x2 + y2 = 4, x2 + y2 = 9,
5.29.		1				5.30.	x = 0, y = 0(x ≤ 0, y ≥ 0);
		2
		2
	μ = 4x +9 y2 .						μ = (y − 2x) (x2 + y2 ).

5.31.D : x = 14 , y = 0, y2 =16x (y ≥ 0);

μ=16x +9 y2 2.

Задача 6. Пластинка D задана неравенствами, μ - поверхностная плотность. Найти массу пластинки.

			D : 1 ≤ x2 9 + y2 4 ≤ 2;
6.1.	D : x2 + y2 4 ≤1;	6.2.	y ≥ 0,	y ≤	2 x;
	μ = y2 .		μ = y	x.	3
			μ = y	x.

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