кратные и криволинейные интегралы

ВАРИАНТ 20

1.

ZZD

3y2

sin

xy

dx dy

2

D : x = 0; y = r

2

4

; y =

x

3

3

2.

ZZD

x + y

dx dy

x2 + y2

x2 + y2 + 2y > 0;

D :

x2

+

y2

y

0;

8

+ 4y 6

< 0 6 x 6

p

3

3.

:

x = p

; x = 2p

;

y

y

x +

1

z = 2; z = 0

3

4.

z = 9

x2 + y2;

z = 22

x2

y2

p

5.

x

16 6 x2 + y2

+ z2 6 100; y

6 p

;

3

z 6 r

; y > xp3

3

x2

+ y2

6.

S :

5x2 = 3y2 + 3z2

(x > 0);

D :

y2

+ z2 6 1; x = 0

4

7.

1 6

x2

+ y2 6 100; x > 0; y >

1

x;

4

2

=

x

y2

8.

L : = a(1 cos '); 0 6 ' 6

(a > 0)

9.

~

2

2 ~

2

2

~

x; 0 6 x 6 1

F = x + y i + x

y j;

L : y = 2 x; 1 < x 6 2 ;

M(2; 0); N(0; 0):

10.

Z

ex [(1 cos y) dx (y sin y) dy];

C

C : fx = 0; y = 1; y = xg

11.

~ ~

F = (3x 3y + 2z) i; p : 3x 2y 2z + 6 = 0

ВАРИАНТ 21

1.

ZZ

8xy + 9x2y2 dx dy

Dp

D : x = 1; y = 3 x; y = x3

2.

ZZ

9 x2 y2 dx dy

D

p

x2 + y2 6 3x;

D : x2 + y2 6 3y

3.

x2 + y2 = 18; y = p

;

3x

z =

25

x; y = 0; z = 0

11

4.

15

17

z =

px2 + y2; z =

x2 y2

2

2

5.

16 6 x2 + y2 + z2 6 81; y 6 0;

z > r			99	; y 6 xp3
		x2	+ y2

6.

S : 2x + 3y + z = 6;

	x2		y2
D :		+		6 1; z = 0
	4		9

7.

x2

p

3

1

6

+ y2

6 4; y 6

x; y 6

0;

4

2

= x2y

8.

p

L

часть параболы

y = 2 x; 0 6 x 6 1:

9.

~

~

F = xj;

L : x2 + y2 = a2; x > 0;

M(0; a); N(0; a):

10.

ZC

(2y + x2) dx + (y2x + y x) dy;

11.

C : x = y2 1; x = 0

~

~

~

~

2 = 0

F = xi + j + (y + z) k; p : x + y + z

ВАРИАНТ 22

1.

2.

3.

ZZ

y2 e xy2 dx dy

D

D : x = 0; y = 1; y = x2

ZZ	9			x2			y2 dx dy
				1
	p	x2	+ y2			6 3x;
D

D : x 6 y 6 x

p p

3 3

p

y = 2 x; x + y = 3; z = 3x; z = 0

4.
z = p25 x2 y2; z = r									99
5.								x2	+ y2
64 6 x2 + y2 + z2 6 144;
z > r						; 0 6 y 6 p3
			x2	63
				+ y2					x

6.

S : x2 + y2 = z2 (z > 0);

D : x2 + y2 6 2x; z = 0

7.

16 6

x2

+

y2

6 100; y > x; x > 0;

3

9

=

x

y

8.

x2

ln x

; 1 6 x 6 2:

L : y =

4

2

9.

~

2

2

~ ~

L :

F =

x

y i + xyj;

отрезок MN; M(1; 1); N(3; 4):

10.

Z

(y x)2 dx (x + y)2 dy;

C

C : fy = cos x; y = 0; =2 6 x 6 =2g

11.

~ ~

F = (2x + 3y 3z) j; p : 2x 3y + 2z 6 = 0

ВАРИАНТ 23

1.

ZZ

24xy + 18x2y2

dx dy

D

D : x = 1; y = x3; y = p3

x

2.

ZZD

(9 + x2

+ y2)2 dx dy

1

D :

y >60; x > 0

6

3x x2 + y2

9;

3.

y = p

; x + y = 6;

3x

z = 4y; z = 0

4.

+p

x

= 60

z = 64 x2 y2; z = 1;

2

y2

(внутри цилиндра)

5.

64 6 x2 + y2 + z2 6 196;

z 6 r

; p3 6 y 6 0

x2

3

+ y2

x

6.

S : x2 + y2 + z2 = 25

(y > 0);

D : x2 + z2 6 16; y = 0

7.

1 6

x2

+

y2

6 4; 0 6 y 6 2x;

3

4

= x

8.

L : y = x3; 0 6 x 6 2:

9.

~

~

2

~

L : y = 2

j;

F = yi

y + x

x

x2;

M(0; 0); N(2; 0):

10.

ZC

(x2 + y2) dx + (y2 x2) dy;

C : nx = R2

y2; x = 0o

11.

p

~

~

~

~

2y 2z + 4 = 0

F = xi + (y z) j + 2k; p : x

ВАРИАНТ 24

1.

ZZ

y2 cos xy dx dy

Dp

D : x = 0; y = ; y = 2x

2.	ZZ	(9 x2			y2)2 dx dy
			108
	D
		3x		x2 + y2				9	;
	D : x 66y						6	4

3.

p

y = 3x; x2 + y2 = 18; y = 0; z = 115 x; z = 0

4.

z = 212 px2 + y2; z = 232 x2 y2

5.

64 6 x2 + y2 + z2 6 196;

rr

	x2	+ y2	6 z 6	x2 + y2
		3		15

6.

S : z = 1 x2 y2;

D : x2 + y2 6 1; z = 0

7.

2

y2

p

1

6 x

+

6 4; 0 6 y 6 3 2x;

6

= y

8.

x = 5(t sin t)

L :

;

0

6

t

6

y = 5(1 cos t)

9.

1

1

F~ =

p

~i p

~j;

x

y

x = r cos t

L : y = r sin t ; x >

0; y > 0;

M(r; 0); N(0; r):

10.

ZC

(y x) dx (x + y) dy;

x2

C : (y = br1

; y = 0); a > 0; b > 0

a2

11.

~ ~

~

F = (2x + 3y + 5z) j + k; p : x + 3y + z 3 = 0

ВАРИАНТ 25								7.
1.								1 6 x2 +		y2	6 4; y 6
ZZ		12xy + 27x2y2				dx dy		= xy2 2
D
D : x = 1; y = x2; y = p3								8.
							x				3
2.			y2					L :	y = 10 sin3 t ;
									x = 10 cos t
		ZZ		dx dy				9.
			x2

p

36x; y 6 0;

0 6 t 6 2

D

~

~

2~

x2 + y2 + 2x

0;

F = 2xyi

y j;

D : x2 + y2 > 2

6

L : отрезок MN;

3.

M(0; 0); N(2; 1):

x + y = 3; y2 = 4x;

10.

4.

z = y; z = 0 (z > 0)

ZC

xy2 dx x2y dy;

z = 10 x2

y2

C :

nx = a y ; x = 0o

p

p

z = 3 x2 + y2;

2

2

11.

5.

~

~

~

F = i + (x z + y) k; p : x + 2y + 2z 4 = 0

16 6 x2 + y2 + z2 6 100;

rr

	x2	+ y2	6 z 6	x2 + y2
		3		35

6.

S : x = z2 y2;

D : y2 + z2 6 4; x = 0

ВАРИАНТ 26

1.

ZZ

y2 sin xy2 dx dy

Dp

D : x = 0; y = ; y = x

2.

ZZD y2 dx dy

x

D :

x > 0

16 6 x2 + y2 6 4p2

y;

3.

y = 3 p

z = 0;pz +

x = 3 3y; x = 2 3y;

4.

z = p9 x2 y2; z = r

80

x2

+ y2

5.

9 6 x2 + y2 + z2 6 81; y 6 0;

0 6 z 6 r		x2	24		; y 6 p3
			+ y2		x

6.

S : 2x + y + 3z = 6;

D :	x2 2
	4 + z9 6 1; y = 0

7.

x2

y2

3p

5

2 6

+

6 5; 0 6 y

6

x;

5

5

3

=

32

xy

63

8.

y = 21

t + 1t

6 6

L :

x = ln t

; 1

t

2

9.

~

~

2~

F = 4xyi

x j;

L : y =

1

x2;

4

M(0; 0); N(2; 1):

10.

ZC

y2 dx x dy;

C : fy = cos x; y = 0; =2 6 x 6 =2g

11.

~

~

~

~

p : 3x + 2y + 3z 6 = 0

F = xi + (z x) j + yk;

ВАРИАНТ 27

1.

ZZ

8xy + 18x2y2 dx dy

Dp

D : x = 1; y = 3 x; y = x2

2. ZZ

9 x2 y2 dx dy

D

x2 + y2 6 2y; D : p x 6 y 6 x 3

3.

p

y = 5x; x2 + y2 = 50 z = 1225x; z = 0 (z > 0)

4.

p

z = 100 x2 y2; z = 6;

x2 + y2 = 51 (внутри цилиндра)

5.

49 6 x2 + y2 + z2 6 169; y > 0;

r

	x2	+ y2	x
			6 z 6 0; y > p
		24
				3

6.

S : x2 + y2 + z2 = 1

(z > 0);

7.

D : x2 + y2 6 x; z = 0

1 6

x2

+ 3y2

6 15; x > 0; y >

1

x;

3

3

=

x

y

8.

x = et(cos t + sin t)

L : y = et(cos t sin t)

; 0 6 t 6

4

9.

~

~

2~

F = 2xyi

x j;

L : x = 2y2;

M(0; 0); N(2; 1):

10.

ZC

ex [(1 + cos y) dx (y + sin y) dy];

C : fy = x; x = 2; y = 1g

11.

~

~ ~

F = i + (5x + 2y + 3z) k; p : x + y + 3z 3 = 0

ВАРИАНТ 28

1.

ZZD

y2 e xy8 dx dy

D : x = 0; y = 4; y = 2x

2.

ZZD

(x y) dx dy

D : x >60; y > 0

6

2x

x2

+ y2

4x;

3.

z = 0; p

= 2

p

+

x = 19 2y; x = 4 2y;

y

z

4.

z =

p

x2

+ y2

36

x2 y2;

z = r

63

5.

x

9 6 x2 + y2 + z2 6 64; y

6 p

;

3

z > r

y 6 p3

99

;

x2 + y2

x

6.

S : x2 + y2 + z2 = 4								(x > 0);
D : z2 + y2 6 2z; x = 0
7.
	x2					y2
1 6				+			6 9; x 6 y 6 0;
		3				1
=			20		xy2
	363
8.								; 0 6 t 6 2
L : y = sin t t cos t
x = cos t + t sin t
9.	~					~		2~
				F = 2yxi y j;

L ломаная линия, первое звено которой соединяет точки M(0; 0) и B(2; 0), а второе точки B(2; 0) и N(2; 1).

10.

	ZC	x dx + 2 ln x dy;
		y
	C : f2x + y = 4; x = 1; y = 0g
11.	~ ~	~
~			4 = 0
F = xi + j + (y + z) k; p : 2x + y + 2z

ВАРИАНТ 29

1.

ZZ

12xy + 27x2y2

dx dy

D

D : x = 1; y = x3; y = p

x

2.

ZZ ln

p

9 x2 y2 dx dy

D

1 6 x2 + y2 6 4;

D : y > x; y > px

3

3.

x =

5

p

; x =

5

y;

y

3

9

z = 0; z =

5

(3 + p

)

y

9

4.

x

+p

= 45

z = 81 x2 y2; z = 5;

2

y2

(внутри цилиндра)

5.

36 6 x2 + y2 + z2 6 100;

z > r			63	; p3 6 y 6 xp3
		x2	+ y2	x

6.

S : x2 + y2 + z2 = 4 (y > 0);

D : x2 + z2 6 2x; y = 0

7.

9 6 x2 +					y2	6 100; 0 6 y 6 3x;
		y		3
=
		x2
8.		x = t3 + 1
	L : y = t2						; 0 6 t 6 1
9.
	~					~	~
	F = cos y i						sin x j;
	L :			отрезок MN;
	M(2; 2); N( 2; 2):
10.
ZC	(x2y + x y) dx + (y2 + 2x) dy;
C : y = x2 + 1; y = 2
11.	~			~		~
~

F = (z + x) i + yj + k; p : 2x + 2y + z 4 = 0