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Национальный авиационный университет

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Evaluate line integrals of the second type.

10. ∫ x2 + 8ydx − (xy + 1)dy, if L is the arc of the parabola y = x2 between

two points А(0; 0) and В(2; 4).

11. ∫ xdy − ydx, if L is the arc of the curve

0 ≤ t ≤ π / 2 .

12. ∫ y cos3 xdx + y−2 dy, if L is the arc of the curve

x = 2cos3 t,

y = tg x , π4

y= 2sin3 t,

≤x ≤ π3 .

13.	∫ y2 dx + x2 dy, if L is the first	arc of the cycloid x = a(t − sin t),
	L
y = a(1− cos t).
14.	∫ xydx + zdy + (x2 + y2 )dz, if L is the arc of the screw line x = a cos t,
	L
y = a sin t, z = bt ( 0 ≤ t ≤ π ).
15.	∫ (x − 1)dx + (x − y)dy + (2z − x)dz,	if L is a line segment connecting the

points А(0; 0; 0) and В(1; 2; 3).

16.	∫ (x2 − y)dx + ( y2 + 2x)dy, if L is a broken line consequently connecting
	L
the points А(0; 0), В(1; 1), С(1; 0) and D(3; 0).
Evaluate line integrals of the second type using Green’s formula.
17.	∫ 2xdy − ydx if L is the closed contour formed by parts of the parabola
y = x2	L
y = x2	and the straight line y = x .
18.	∫ (1− x2 ) ydx + (1+ y2 )xdy if L is the circumference x2 + y2 = R2 .
	L
19.	∫ (xy + x + y)dx + (xy + x − y)dy if L is the circumference x2 + y2 = 2x.
	L
20.	∫ xydx + (x2 + y2 )dy if L is a contour of the triangle whose vertices are
	L
points А(1; 0), В(2; 1) and С(1; 2).
21.	Determine coordinates of the gravity center of the homogeneous arc of
the cycloid x = a(t − sin t), y = a(1− cos t), 0 ≤ t ≤ 2π.
22.	Evaluate the work done by the force F = {x; y} at moving a material

point along the following curve y = t cos t − sin t, x = t sin t + cost, t [0; π / 2] .

23. Evaluate the work done by the force F = {yx; yz; xz} at moving a material point along the line segment connecting the points А(0; 1; 1) and В(2; –1; 3).

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Evaluate line integrals making sure, that they do not depend on the form of integration path.

	(2;0)
24.	∫	(2y2 − 3x2 y)dx + (4xy − x3 )dy.
	(1;1)
	(0; 2; 3)
25.	∫	(2x − y3 )dx + (z2 − 3xy2 )dy + 2yzdz.
	(1; 0; 0)

26.Evaluate the area bounded by the astroid whose parametric equations are

x= a cos3 t, y = a sin3 t, using the line integral.

27.Evaluate the mass of the curve x2 + y2 = R2 ( x ≥ 0 , y ≥ 0 ) whose linear density is γ(x; y) = x.

Answers

	1.	7 / 24. 2. 256/15.	3.	5 ln 2.	4.	(17	17 − 2 2) / 6.				5.	8.	6.		4/ 3.	7. R4 / 3.
8.	2π	1 + b2 (3 + 4π2b2 ). 9.	7	26. 10. −	54	. 11.	−	3π	. 12.	23 + 2	2	− 8	3	.	13.	πa3(5 − 2π).
	3		2		5			4			24

14. πa2b − 2. 15. 6. 16. 7,5. 17. 0,5. 18. πR24 . 19. −π. 20. 43 . 21. (π; 4a / 3). 22. π2 /8.

23. 14/3. 24. –1. 25. 17. 26. 3πa2 . 27. R2. 8

Micromodule 7

SELF-TEST ASSIGNMENTS

7.1. Evaluate line integrals of the first type.

7.1.1. ∫ x2 + y2 + z2 dl where L is the arc of the screw line x = 3cost,

y = 3sin t, z = 4t, 0 ≤ t ≤ 2π.

7.1.2. ∫ x2 ydl	where L is the circumference arc x2 + y2	= 4,		x ≥ 0,	y ≥ 0.
L			1			1
7.1.3. ∫ xyzdl,	where L is the arc of the curve x = t, y =		1	8t3 , z =		1	t2	,
7.1.3. ∫ xyzdl,	where L is the arc of the curve x = t, y =			8t3 , z =			t2	,
L		3			2
0 ≤ t ≤ 1.
7.1.4. ∫ 2zdl where L is the arc of the curve x = et cos t,		y = et sin t,			z = et ,
0 ≤ t ≤ 2π.L
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7.1.5. ∫

dl where L is the arc of the screw line

x = cos t,

y = sin t,

+ y

z = t, 0 ≤ t

≤ 2π.

7.1.6. ∫ (2z −

x2 + y2 )dl where L is the arc of the screw line x = 2cost,

y = 2sin t,

z = t, 0 ≤ t ≤ π.

7.1.7.

∫

where L is the arc of the circumference

x = cos t,

− y

)

y = sin t,

0 ≤ t ≤

π .

7.1.8.

∫ (2x + 3y − z)dl,

if L is the line segment connecting the points А(3;

–1; 6) and В (1; 0; 4).

7.1.9.

∫ dl

where L is the arc of the curve

x = 16t, y =

t3 ,

z =

t5 ,

t [0; 2].

7.1.10. ∫ ydl

where L is the parabola arc of

y2 = 4x

which is enclosed in

the parabola x2

= 4 y.

7.1.11. ∫ dl

where L is the arc of the curve x = 2(t − sin t ) ,

y = 2(1− cos t ),

0 ≤ t ≤ π .

∫ (x2 + y2 )dl where L is the arc of the curve

7.1.12.

x = cos t,

y = sin t,

z = t, 0 ≤ t ≤ 2π.

where L is the arc of the curve x = 5(t − sin t ),

y = 5(1− cos t ),

7.1.13. ∫ ydl

0 ≤ t ≤ 2π. L

7.1.14. ∫ ydl

where L is the arc of the curve x = 4cos3 t,

y = sin3 t,

0 ≤ t ≤ π .

7.1.15.

∫ x2 dl

where L is the arc of the curve y = ln x

from

the

point

x1 =

x2 = 2 2.

3 to the point

7.1.16.

∫ xy2 dl

where L is the quarter of the circumference x2 + y2

= 4,

x ≥ 0,

y ≤ 0.

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7.1.17.

∫

x2 + y2 dl

where

L is

part

the

curve

x = cos t + t sin t,

0 ≤ t ≤ 2π .

y = sin t − t cost,

7.1.18.

∫

x2 + y2 dl

where

L is

part

the

circumference

x = cos t,

π .

y = sin t, 0 ≤ t ≤

7.1.19. ∫ x2dl

where L is the arc of the curve

x = cost + t sint,

y = sint − t cost,

0 ≤ t ≤ π .

7.1.20. ∫ ydl

where L is the arc of the previous variant.

7.1.21. ∫

x2 + y2 dl

where L is a quarter of the circumference

x2 + y2 = 9,

x ≥ 0, y ≥

7.1.22.

∫ x2 dl

where L is a

quarter

the

circumference

x2 + y2 = 25,

x ≤ 0, y ≥

7.1.23. ∫ (x2 + yz)dl

if L is a line segment connecting two points А(1; –2; 3)

and В(5; 0; 2).

where L is the arc of the curve x = 3(t − sin t),

7.1.24. ∫ xdl

0 ≤ t ≤ 2π. .

y = 3(1− cos t),

x = 3cos3 t,

7.1.25. ∫ xdl

where L is the arc of the curve y = 3sin3 t,

≤ t

≤

2 .

7.1.26. ∫ − ydl

where L is the arc of the curve x = cos3 t,

y = sin3 t,

0 ≤ t ≤ π .

7.1.27.

∫ xydl,

where L is the arc of the screw line

x = cos t,

y = sin t,

z = t, t [0; 2π].

7.1.28.

∫ xyzdl

where L is the arc of the screw line

x = cos t,

y = sin t,

z = 3t, 0 ≤ t ≤ 2π .

7.1.29. ∫ ydl

where L is the arc of the curve x = cos3 t, y = sin3 t,

0 ≤ t ≤ π .

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7.1.30. ∫ xydl where L is a part of the circumference x = 6 cos t, y = 6 sin t,

0 ≤ t ≤ 32π .

7.2. Evaluate line integrals of the second type (integration should be fulfilled in a positive direction).

7.2.1. ∫ 2ydx − (3y + x2 )dy		where L is the arc of the parabola y = x2 − 4x ,
L
lying under the axis Ox .
7.2.2. ∫ x2 ydx + x3dy where L is the parabola arc y2						= x,	0 ≤ x ≤ 4,	y ≥ 0.
L
7.2.3.	∫ xydx + yzdy + z2 xdz		where	L is	a quarter	of	the circumference
	L	0 ≤ t ≤ π .
x = 2 cos t,	y = 1, z = 2 sin t,	0 ≤ t ≤ π .
			2
7.2.4. ∫ x4 ydy − y4 xdx where L is the arc of the curve						x =	cos t , y = sin t ,
L
t [0; π / 2] .
7.2.5.	∫(x2 − y2 )dy where		L is the	arc	of the cubic		parabola	y = 2x3 ,
0 ≤ x ≤ 1.	L

7.2.6. ∫(x2 + y2 )dx where L is the arc of the parabola					y = 2x2 ,		2 ≤ x ≤ 4.
L
7.2.7. ∫ x2 dx + x ydy if L is a quarter of the circumference							x2 + y2 = 25,
L
x ≥ 0, y ≥	0 , direction of integration is counter-clockwise.
7.2.8.	∫(x − y)dx + (x + y)dy	where L is the arc			of the cubic parabola
	L
y = 2x3 , 0 ≤ x ≤ 1.
7.2.9.	∫ (x − y)dx + (x + y)dy	if AB is a line segment connecting the points
AB
A (2; 3) and B (3; 5).
7.2.10. ∫(x2 − y3 )dx + (x + y)dy where			L	is the	arc	of	the	parabola
	L
x = y2 − 1, 0 ≤ y ≤ 1.
7.2.11.	∫ x1/ 3 ydy − y1/ 3 xdx where L is		the	arc of	the	curve		x = cos3 t ,
y = sin3 t ,	L
y = sin3 t ,	t [0; π / 2] .
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7.2.12. ∫(6xy − 1)dx + 2 y2 xdy where L is the parabola arc x = 3y2 ,				y [0; 1].
	L
7.2.13.	∫(2x2 y − y2 )dx + 6xydy		where L is the arc of the cubic parabola
	L
y = 2x3 , 0 ≤ x ≤ 1.
7.2.14. ∫ 2ydx − ( y − x2 )dy		where L is the parabola arc y = x − x2 ,		x [0; 1] .
	L
7.2.15.	∫ 2xydx + 3xy2dy	if AВ is a segment of the straight line connecting
	AB
the points A (1; 1) and B (2; 4).
7.2.16.	∫ (x − y2 )dx + (x + y)dy		if AВ is a line segment connecting the
	AB

points А(0; 0) and В(1; 2).

7.2.17. ∫ (6xy2 + 4x3 )dx + (6x2 y + 3y2 )dy if AВ is a line segment connec-

ting the points А(2; 3) and В(3; 4).

7.2.18. ∫ (2xy − 5y3 )dx + (x2 −15xy2 + 6y)dy if АВ is a line segment con-

necting the points А(0; 0) and В(2; 2).

7.2.19. ∫	xdx + (2x + y)dy		if АВ is a line segment connecting the points
	(x + y)	2
AB

А(1; 1) and В(3; 2).

7.2.20. ∫ (2xy2 + 3x2 )dx + (2x2 y + 3y2 )dy if AВ is a line segment connecting

the points А(1; 2) and В(2; 1).

7.2.21.	∫	− y2 dx + x2dy		if AВ is a line segment connecting the points А(2; 1)
7.2.21.	∫	(x − y)	2	if AВ is a line segment connecting the points А(2; 1)
	AB	(x − y)
and В(5; 3).
7.2.22.	∫ ydx − (x2 + y)dy where L is the arc of the parabola						y = 2x − x2 ,
	L
lying above the axis Ox.
7.2.23.	∫ (3y2 + 4 y)dx + (6xy + 4x − 4 y)dy if AВ is a line segment connec-
	AB
ting the points А(0; 1) and В(2; 5).
7.2.24. ∫(5x − 2y)dx + x2 ydy where L is the arc y =					1	x3 , 0 ≤ x ≤ 1.
7.2.24. ∫(5x − 2y)dx + x2 ydy where L is the arc y =						x3 , 0 ≤ x ≤ 1.
	L			3
7.2.25. ∫(4xy − y2 )dx + 2xydy where L is the arc of the parabola							y = 2x2 − 4x,
0 ≤ x ≤ 2.	L
0 ≤ x ≤ 2.
176

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7.2.26. ∫3x3 ydx + xydy where L is the parabola arc y = 2x2 + 6x, 0 ≤ x ≤ 3.

7.2.27. ∫ 4xydx + 3xydy + zdz if AВ is a line segment connecting points А(0; 1; 5)

and В(2; 1; 3).
7.2.28.	∫ (2zy − 3y2 )dx + 6ydy + xdz			if	AВ is		a	line	segment connecting
	AB
points А(1; –1; 0) and В(2; 3; 7).
7.2.29.	∫(x2 y + 2 y2 )dx − 2xydy,		if	L	is	an	arc	of	the	cubic parabola
	L
y = (x − 1)3 , 0 ≤ x ≤ 1.
7.2.30.	∫ (2x2 − 6z)dx + x2 ydy + (2z − 1)dz,					if	AВ is		a	segment of the
	AB
straight line connecting the points А(–1; 0; 4) and В(1; 4; 2).
7.3. Evaluate a line integral		∫ P(x, y)dx + Q(x, y)dy over a closed contour L
		L
using Green’s formula. Direction of integration is counter-clockwise.
7.3.1.	∫(2x + y)dx − (3y + 2x)dy		if	L is a		contour of			the triangle whose
	L
vertices are А(1; 2), В(3; 1) , С(2; 5).
7.3.2.	∫ (x + y2 )dx + 4xdy	if L is		a contour			of	the	rectangle 1 ≤ x ≤ 3 ,
0 ≤ y ≤ 4 .	L
0 ≤ y ≤ 4 .
7.3.3.	∫ ydx + x2dy if L is a contour formed by the parabola y = x2 and the
	L
straight line. y = 2x + 3 .

7.3.4. ∫ y2dx + xydy if L is a contour of the rectangle −1 ≤ x ≤ 1, 0 ≤ y ≤ 3 .

	L
7.3.5. ∫ x2 y2 dx + xydy L is a contour formed by the parabolas y = x2					and y2 = x.
	L
7.3.6.	∫ y2 dx + x2 ydy if	L is	a	contour of the rectangle	−1 ≤ x ≤ 4 ,
1 ≤ y ≤ 3.	L
1 ≤ y ≤ 3.
7.3.7.	∫ x3dx + (x + 2 y)dy	if L is a contour formed by the parabola				y = x2
	L
and the straight line y = 3x + 4.
7.3.8.	∫ (2x − y)dx + (4 y + x)dy,		if	L is a contour of the triangle		whose
	L
vertices are А(0; 4), В(4; 0) , С(2; –2).
						177

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7.3.9. ∫ (x2 + y)dx + (3x − y)dy if L is a contour formed by the parabola

y = x2 − 1 and the straight line y = 2x + 2 .

7.3.10. ∫(2x − y2 )dx + ( y + x2 )dy if L is a contour of the rectangle 0 ≤ x ≤ 2 ,

1 ≤ y ≤ 4 .

7.3.11. ∫ (x − y)dx + ( y + x)dy if L is a contour of the triangle whose vertices

are А(0; 4), В(4; 0) , С(2; -2).

7.3.12. ∫ x2 ydx − xy2dy if L is the circumference x2 + y2 = 4.

	L
7.3.13.	∫ (xy − x2 )dx + xdy if L is a contour formed by the parabola y = x2
	L
and the straight line y = x + 2 .
7.3.14.	∫ ydx + (x2 + y2 )dy		if	L is a contour of the rectangle 0 ≤ x ≤ 4 ,
1 ≤ y ≤ 2.	L
1 ≤ y ≤ 2.
7.3.15. ∫ (2x + y)dx + 2ydy			if L is a contour of the triangle whose vertices
	L
are А(–2; 0), В(0; 3), С(2; 1).
7.3.16.	∫(x2 + 2y)dx + ydy			if L is a contour formed by the parabolas
	L
y = 2x − x2	and y = x2 + x − 1 .
7.3.17. ∫ − x2 ydx + xy2 dy if L is the circumference x2 + y2 = 9 .
	L
7.3.18.	∫ (x2 − y)dx + (x − y)dy			if L is a contour formed by the parabola
	L
y = 6x − x2	and the straight line		y = 5.
7.3.19.	∫ (x2 + 2y2 )dx + 2ydy if L is a contour of the rectangle −2 ≤ x ≤ 0 ,
1 ≤ y ≤ 2 .	L
1 ≤ y ≤ 2 .
7.3.20.	∫ ( y2 − x2 )dx + x2 dy		if	L is a contour of the rectangle 0 ≤ x ≤ 1,
−1 ≤ y ≤ 2 .	L
−1 ≤ y ≤ 2 .
7.3.21. ∫ y2 dx − xydy		if L is a contour of the triangle whose vertices are А(1; 2),
	L
В(3; 1), С(2; 5).
7.3.22. ∫ y2 dx + x2 dy		if L is a contour of the rectangle 1 ≤ x ≤ 2 , 0 ≤ y ≤ 2 .
	L
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7.3.23. ∫3ydx − (2y + x2 )dy if L is a contour formed by the parabola y = 4x − x2

and the straight line y = 3 .

7.3.24. ∫ xydx + x2 dy if L is the circumference x2 + y2 = 25 .

	L
7.3.25. ∫ (x + 4y)dx − ydy		if L is a contour of the triangle whose vertices are
	L
А(1; 2), В(3; 1), С(2; 5).
7.3.26.	∫ (2x + y2 )dx + ( y − x)dy if L is a contour of the triangle		whose
	L
vertices are А(–3; 1), В(4; 1) , С(2; 3).
7.3.27.	∫ (x + y)dx + x2 dy	if L is a contour formed by the parabolas	y = x2
	L
and y = 2 − x2 .
7.3.28.	∫ (x + 3y)dx + x3dy if L is a contour of the rectangle −3 ≤ x ≤ 0 ,
−1 ≤ y ≤ 0 .	L
−1 ≤ y ≤ 0 .
7.3.29. ∫ xydx + (x + 1)dy		if L is the circumference x2 + y2 = 16 .
	L

7.3.30. ∫ (2x + y)dx + ( y − 3x)dy if L is a contour of the triangle whose vertices

are А(–2; 0), В(0; 3) , С(2; 1).

7.4. Verify that the given line integral is path-independent and evaluate it.

	(3; 4)
7.4.1.	∫	( y4 + 2xy)dx + (4xy3 + x2 − 3y2 )dy.
	(1;1)
	(2;3)
7.4.2.	∫	(2xy + 3y − 8x)dx + (x2 + 3x)dy .
	(1; 0)
	(1; 0; 4)
7.4.3.	∫	x( y2 + z2 )dx + y(x2 + z2 )dy + z(x2 + y2 )dz.
	(0; 0; 1)
	(2; 4)
7.4.4.	∫	(3x2 y4 − 4)dx + (4x3 y3 + 3y2 )dy .
	(−4; 0)
	(3; 2; 0)
7.4.5.	∫	( y2 z3 + 2)dx + (2xyz3 + 1)dy + (3xy2 z2 + 2z)dz .

(0; 0; 0)

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	(1;3)
7.4.6.	∫	( y3 + 2 y − 3x2 )dx + (3xy2 + 2x)dy .
(−1; 0)
(3; −1)
7.4.7.	∫	(2xy6 − 6x)dx + (6x2 y5 + 4y)dy .
	(0; 5)
	(0;3)
7.4.8.	∫	(4x3 y3 + 1)dx + (3x4 y2 − 4y3 )dy .
(−3; −1)
(0;3; 2)
7.4.9.	∫	( y + z + yz)dx + (x − z + xz)dy + (x − y + xy)dz.
(1; 0; 0)
	(3;5)
7.4.10.	∫	(5x4 y3 + 9x2 )dx + (3x5 y2 + 2 y)dy .
	(0; 0)
	(3;3;3)
7.4.11.	∫	(3x2 y2 z − 1)dx + (2x3 yz − 2)dy + (x3 y2 − 3)dz .
	(0;1; 2)
	(1;3)
7.4.12.	∫	(3x2 y + y2 + 2x)dx + (x3 + 2xy)dy .
	(−2; −1)
	(1; 0; 4)
7.4.13.	∫	( yz + 2xy2 z2 )dx + (xz + 2yx2 z2 )dy + (xy + 2zx2 y2 )dz.
	(0; 2; 0)
	(3;3;5)
7.4.14.	∫	(x − yz)dx + (2 y − xz)dy + (2z − xy)dz .
	(1; 0; 0)
	(3; 4)
7.4.15.	∫	(3x2 y + 2xy4 + 2x)dx + (x3 + 4x2 y3 )dy.
	(0; −2)
	(1; 4; 2)
7.4.16.	∫	( yz2 + 1)dx + (xz2 + 2y)dy + (2xyz + 3z2 )dz .
	(0; 0; 0)
	(1; 4)
7.4.17.	∫	(4x3 y2 + 3x2 y + 2x)dx + (2x4 y + x3 + 2y)dy .
	(0; 0)
	(1; 2; 3)
7.4.18.	∫	(2xy + z2 + 3)dx + (x2 + 2 yz − 2y)dy + (2xz + y2 )dz.
	(0; 0; 0)
	(2; 4)
7.4.19.	∫	(2x3 − 3y2 + 4y)dx + (4x − 6xy − 2y)dy.
	(1;1)

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