9

D: y =

p

, y = p, x =

1

, x = 1

D: x = 0, y =

y = 2x

2p ,

2

2

4.11

òò12 y sin 2xydxdy

4.12 òò y 2 cos 2xydxdy

D

p

p

D

x

D: y =

, y =

, x = 2, x = 3

D: x = 0, y =

y =

p / 2,

4

2

2

4.13

òò ye xy / 4 dxdy

4.14 òò3y 2 sin

xy

dxdy

D

D

2

2

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

D: x = 0, y =

y =

x

4p / 3,

3

4.15

òò2 y cos 2xydxdy

4.16 òò y 2 cos xydxdy

D

p

p

D

D: y =

, y =

, x = 1, x = 2

D: x = 0, y =

y = x

p ,

4

2

4.17

òò y sin xydxdy

4.18 òò y 2 sin

xy

dxdy

2

D

1

D

D: y = p, y = 2p , x =

, x = 1

D: x = 0, y =

y = x

p ,

4.19

òò8 ye4 xy dxdy

2

4.20 òò y cos xydxdy

D

1

1

D

1

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

, x =

D: y = p, y = 3p , x =

, x = 1

2

4

2

4.21

òò6 ye xy / 3dxdy

4.22 òò y sin 2xydxdy

D

D

1

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

D: y = p / 2, y = 3p / 2, x =

, x = 2

2

4.23

òò y cos 2xydxdy

4.24 òò3y sin xydxdy

D

1

D

D: y = p / 2, y = 3p / 2, x =

, x = 2

D: y = p / 2, y = 3p , x = 1, x = 3

2

4.25

òò12 ye6 xy dxdy

D

1

1

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

, x =

3

6

Задача 5. Вычислить тройной интеграл.

5.1 òòò2 y

2 e xy dxdydz

5.2 òòò y 2 cos(

pxy

)dxdydz

V

V

2

10

V:

x = 0, y = 1, y = x

V:

x = 0, y = -1, y = x

z = 0, z = 1

z = 0, z = 2p 2

5.3 òòò y 2 z cos(

xyz

)dxdydz

5.4 òòò x 2 z sin( xyz)dxdydz

V

3

V

V:

x = 3, y = 1, z = 2p

V:

x = 2, y = p, z = 1

x = 0, y = 0, z = 0

x = 0, y = 0, z = 0

5.5 òòò z 2 y cos(

xyz

)dxdydz

5.6 òòò x 2 sin(

pxy

)dxdydz

2

V

3

V

V:

x = 0, y = 0, z = 0

V:

x = 2, y = x, y = 0

x = 3, y = p, z = 1

z = 0, z = p

5.7 òòò y 2 e xy / 2 dxdydz

5.8 òòò y 2 cos(pxy)dxdydz

V

V

V:

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

V:

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

z = 0, z = -1

z = 0, z = p 2

5.9 òòò y 2 z cos(

xyz

)dxdydz

5.10 òòò x 2 z sin(

xyz

)dxdydz

V

9

V

4

V:

x = 9, y = 1, z = 2p

V:

x = 1, y = 2p , z = 4

x = 0, y = 0, z = 0

x = 0, y = 0, z = 0

5.11 òòò8 y 2 ze - xyz dxdydz

5.12 òòò z 2 cos(

pyz

)dxdydz

V

V

2

V:

x = 2, y = -1, z = 2

V:

y = 0, z = -1, y = z

x = 0, y = 0, z = 0

x = 0, x = 4p 3

5.13 òòò2x 2 z sin( 2xyz)dxdydz

5.14 òòò y 2 z cos(

xyz

)dxdydz

V

V

2

V:

x = 2p , y =

1

, z =

1

V:

x = 0, y =

p

, z = 0

2

2

2

x = 0, y = 0, z = 0

x = 2, y = 0, z = -1

5.15 òòò8 y 2 ze 2 xyz dxdydz

5.16 òòò y 2 sin(

pxy

)dxdydz

V

V

2

V:

x = -1, y = 2, z = 1

V:

x = 0, y = -1, y = x

x = 0, y = 0, z = 0

z = 0, z = 2p 2

5.17 òòò x 2 sin(pxy)dxdydz

5.18 òòò y 2 e- xy dxdydz

V

V

V:

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

V:

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

11

z = 0, z = 4p

z = 0, z = 1

5.19 òòò y 2 cos(

pxy

)dxdydz

5.20 òòò x 2 z sin(

xyz

)dxdydz

V

4

V

2

V:

x = 0, y = -1, y =

x

V:

x = 1, y = 4, z = p

2

z = 0, z = -p 2

x = 0, y = 0, z = 0

5.21 òòò z 2 sin(

pxz

)dxdydz

5.22 òòò2x 2 z sin( 2xyz)dxdydz

V

2

V

V:

x = -1, x = z, y = 0

V:

x = 2, y =

1

, z = p

2

y = p, z = 0

x = 0, y = 0, z = 0

5.23 òòò y 2 e- xy dxdydz

5.24 òòò2 y 2 ze xyz dxdydz

V

V

V:

x = 0, y = -1, y = x

V:

x = 1, y = 1, z = 1

z = 0, z = 2

x = 0, y = 0, z = 0

5.25 òòò x 2 sin( 4pxy)dxdydz

V

V:

x = 1, y =

x

, y = 0

2

z = 0, z = 8p

Задача 6. Вычислить объем тела, ограниченного поверхностями.

6.1	z = 9 - y 2 , 3x + 4 y = 12, z = 0,							x = 0, y ³ 0
6.2	z = x 2	+ y 2 ,	z = x 2 + 2 y 2 ,				y = x,		y = 2x, x = 1
6.3	x = 4 - y 2 ,		x = 2 + y 2 , z = -1,					z = 2
6.4	z = y 2 , 2x + 3y = 6,					x = 0,	z = 0
6.5	z = y 2	+1, x + y = 1,				z = 0,	y = 0,		x = 0
6.6	z = x 2	+ y 2 ,	y = x 2 ,			y = 1,	z = 0
6.7	y 2 = 4x + 4,		y 2 = -2x + 4,				z = 0,		z = 3
6.8	z = x 2 , 2x - y = 0,					x + y = 9,		z = 0
6.9	z = 2 - x, y = 2					y = z = 0
				x,

6.10	x 2 + y 2			= 1,		z = 2 - x - y, z = 0
6.11	z =					y = x 2 ,			z = 0
		1 - y,
6.12	y =			y = 2					x + z = 6, z = 0
			x,				x,
6.13	z = y 2 ,			y = 2x,			x = 3, z = 0
6.14						z = 2 y, y = 0
	z =	9 - x 2 ,

12

6.15y = 9 - x 2 , y + z = 3, z = 0, y = 0

6.16y = 3x 2 , y = 7, z = y, z = 0

6.17

y = z 2 ,

2x - z = 0, x + z = 6,

y = 0

6.18

y = x 2 ,

x = z,

x + z = 4,

y = 0

6.19

z =

z = 2

x + y = 4,

y = 0

x,

x,

6.20

x = 4 - y 2 ,

z + y = 2, x = 0,

y = 0,

z = 0

6.21

y = x 2 - 4,

y = 7, z = y, z = 0

6.22

x 2 + y 2

= R 2 ,

z ³ 0, z = y, x =

R

6.23

2

z = x2 + y2 ,

x - y = 0,

3

x - y = 0,

x + y = 2

6.24

z = x 2 + 3y 2 ,

x + y = 1,

x = 0, y = 0, z = 0

6.25

x 2 + y 2

- z = 0, x = 2, y = 3,

x = 0,

y = 0, z = 0

Задача 7. Тело V задано ограничивающими его поверхностями,

m - плотность. Найти массу тела.

7.1 64(x 2 + y 2 ) = z 2 , x 2 + y 2			= 4,	7.2 x 2 + y 2 = 1, x 2 + y 2 = 2z,
y = 0, z = 0 ( y ³ 0, z ³ 0)				x = 0, y = 0, z = 0 (x ³ 0, y ³ 0)
m = 5(x 2 + y 2 ) / 4				m = 10x
7.3 x 2 + y 2 = 16z 2 / 49, x 2 + y 2 = 4z / 7,				7.4 x 2 + y 2 + z 2 = 1, x 2 + y 2 = 4z 2 ,
x = 0, y = 0 (x ³ 0, y ³ 0)				x = 0, y = 0 (x ³ 0, y ³ 0, z ³ 0)
m = 80 yz				m = 20z
7.5 36(x 2 + y 2 ) = z 2 , x 2 + y 2			= 1,	7.6 x 2 + y 2 = 4z 2 / 25, x 2 + y 2 = 2z / 5,
x = 0, z = 0 (x ³ 0, z ³ 0)				x = 0, y = 0 (x ³ 0, y ³ 0)
m =	5	(x2 + y 2 )		m = 28xz
6
7.7 x 2 + y 2 + z 2 = 4, x 2 + y 2 = z 2 ,				7.8 25(x 2 + y 2 ) = z 2 , x 2 + y 2 = 4,
x = 0, y = 0 (x ³ 0, y ³ 0, z ³ 0)				x = 0, y = 0, z = 0 (x ³ 0, y ³ 0, z ³ 0)
m = 6z				m = 2(x 2 + y 2 )
7.9 x 2 + y 2 = 1, x 2 + y 2 = 6z,				7.10 x 2 + y 2 = z 2 / 25, x 2 + y 2	= z / 5,
x = 0, y = 0, z = 0 (x ³ 0, y ³ 0)				x = 0, y = 0 (x ³ 0, y ³ 0)
m = 90 y				m = 14 yz
7.11 x 2 + y 2 + z 2 = 4, x 2 + y 2			= 9z 2 ,	7.12 9(x 2 + y 2 ) = z 2 , x 2 + y 2 = 4,
x = 0, y = 0 (x ³ 0, y ³ 0, z ³ 0)				x = 0, y = 0, z = 0 (x ³ 0, y ³ 0, z ³ 0)
m = 10z				m = 5(x 2 + y 2 ) / 3
7.13 x 2 + y 2 = 1, x 2 + y 2 = z,				7.14 x 2 + y 2 = z 2 / 49, x 2 + y 2	= z / 7,
x = 0, y = 0, z = 0 (x ³ 0, y ³ 0)				x = 0, y = 0 (x ³ 0, y ³ 0)
m = 10 y				m = 10xz
7.15 x 2 + y 2 + z 2 = 4, x 2 + y 2			= 4z 2 ,	7.16 16(x 2 + y 2 ) = z 2 , x 2 + y 2	= 1,
x = 0, y = 0 (x ³ 0, y ³ 0, z ³ 0)				x = 0, y = 0, z = 0 (x ³ 0, y ³ 0, z ³ 0)
m = 10z				m = 5(x 2 + y 2 )

				13
7.17	x 2 + y 2	= 4, x 2 + y 2	= 4z,	7.18	x 2 + y 2	= z 2 , x 2 + y 2 = z,
	x = 0, y = 0, z = 0 (x ³ 0, y ³ 0)				x = 0, y = 0 (x ³ 0, y ³ 0)
	m = 5y				m = 35yz
7.19	x 2 + y 2 + z 2 = 1, x 2		+ y 2 = z 2 ,	7.20	x 2 + y 2	= z 2 , x 2 + y 2 = 4,
	x = 0, y = 0 (x ³ 0, y ³ 0, z ³ 0)				x = 0, y = 0, z = 0 (x ³ 0, y ³ 0, z ³ 0)
	m = 32z				m = 5(x 2 + y 2 ) / 2
7.21	x 2 + y 2 + z 2 = 9, x 2		+ y 2 = 4,	7.22	x 2 + y 2	= 1,	x 2 + y 2	= 3z,
	(x 2 + y 2	£ 4), z = 0	(z ³ 0)		x = 0,	y = 0,	z = 0	(x ³ 0, y ³ 0)
	m = 2z				m = 15x
7.23	x 2 + y 2	= 4z 2 / 49, x 2 + y 2 = 2z / 7,		7.24	x 2 + y 2	+ z 2	= 16, x 2 + y 2 = 9z 2 ,
	x = 0, y = 0 (x ³ 0, y ³ 0)				x = 0, y = 0 (x ³ 0, y ³ 0, z ³ 0)
	m = 20xz				m = 5z
7.25	4(x 2 + y 2 ) = z 2 , x 2 + y 2 = 1,
	y = 0, z = 0 ( y ³ 0, z ³ 0)
	m = 10(x 2 + y 2 )

Задача 8. Вычислить криволинейный интеграл

по линии L от точки А до точки В.

8.1 ò(x 2 + y 2 )dx + (x 2 - y 2 )dy, L: y= x , A (-1,1), В (2,2).

L

8.2 ò x ×ex2 dy + ydx , L: y=x , A(0,0) , B (1,1)

L

8.3 ò

L

8.4 ò

xdy - ydx	ìx = a(t - sin t)
	, L: í	, A (2 p a, 0), B (0,0)
	îy = a(1 - cos t)
(x + y)dx + (x - y)dy , L: x 2		+ y 2 = 16 , ( y ³ 0 ) , A(4,0) , B(-4, 0)

	L
8.5	ò(x 2 + y 2 )dx + ( y 2 + x)dy , L: ломаная, состоящая из отрезков x=1, y=5 , A(1,2), B (3,5)
	L
8.6	ò xdy - ydx , L:ломаная ABC , A(-2,0), B(0,2), C (2,0)
	L
8.7	ò xe x3 dy + ydx , L: y = x 2 , A (0,0) , В (1,1)
	L
8.8	ò2xydx - x 2 dy , L: x = 2 y 2	, A (0,0) , B (2,1)
	L
8.9	ò(x + y)dx + (2x - y)dy , L: x 2 + y 2 = 25 , (y ³ 0 ) , A (5, 0) , В (-5,0)
	L
8.10 ò(x 2 - 2xy)dx + (2xy + y 2 )dy , L: y= x 2 , A(1,1), В (2,4)
	L
8.11 ò(3x - y)dy + 2 ydx , L :y=				, A(1,1) , B (4,2)
			x

L