Higher Mathematics. Part 3
.pdfEvaluate line integrals making sure, that they do not depend on the form of integration path.
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(2;0) |
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24. |
∫ |
(2y2 − 3x2 y)dx + (4xy − x3 )dy. |
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(1;1) |
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(0; 2; 3) |
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25. |
∫ |
(2x − y3 )dx + (z2 − 3xy2 )dy + 2yzdz. |
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(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
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7 / 24. 2. 256/15. |
3. |
5 ln 2. |
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(17 |
17 − 2 2) / 6. |
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8. |
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4/ 3. |
7. R4 / 3. |
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8. |
2π |
1 + b2 (3 + 4π2b2 ). 9. |
7 |
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26. 10. − |
54 |
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3π |
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23 + 2 |
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− 8 |
3 |
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13. |
πa3(5 − 2π). |
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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,
L
y = 3sin t, z = 4t, 0 ≤ t ≤ 2π.
7.1.2. ∫ x2 ydl |
where L is the circumference arc x2 + y2 |
= 4, |
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x ≥ 0, |
y ≥ 0. |
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7.1.3. ∫ xyzdl, |
where L is the arc of the curve x = t, y = |
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8t3 , z = |
t2 |
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0 ≤ t ≤ 1. |
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7.1.4. ∫ 2zdl where L is the arc of the curve x = et cos t, |
y = et sin t, |
z = et , |
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0 ≤ t ≤ 2π.L |
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7.1.5. ∫ |
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z3 |
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dl where L is the arc of the screw line |
x = cos t, |
y = sin t, |
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+ y |
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z = t, 0 ≤ t |
≤ 2π. |
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7.1.6. ∫ (2z − |
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x2 + y2 )dl where L is the arc of the screw line x = 2cost, |
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y = 2sin t, |
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z = t, 0 ≤ t ≤ π. |
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7.1.7. |
∫ |
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dl |
where L is the arc of the circumference |
x = cos t, |
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y = sin t, |
0 ≤ t ≤ |
π . |
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7.1.8. |
∫ (2x + 3y − z)dl, |
if L is the line segment connecting the points А(3; |
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–1; 6) and В (1; 0; 4). |
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7.1.9. |
∫ dl |
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where L is the arc of the curve |
x = 16t, y = |
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t3 , |
z = |
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t5 , |
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t [0; 2]. |
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7.1.10. ∫ ydl |
where L is the parabola arc of |
y2 = 4x |
which is enclosed in |
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the parabola x2 |
= 4 y. |
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7.1.11. ∫ dl |
where L is the arc of the curve x = 2(t − sin t ) , |
y = 2(1− cos t ), |
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0 ≤ t ≤ π . |
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2 |
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∫ (x2 + y2 )dl where L is the arc of the curve |
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7.1.12. |
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x = cos t, |
y = sin t, |
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z = t, 0 ≤ t ≤ 2π. |
where L is the arc of the curve x = 5(t − sin t ), |
y = 5(1− cos t ), |
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7.1.13. ∫ ydl |
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0 ≤ t ≤ 2π. L |
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7.1.14. ∫ ydl |
where L is the arc of the curve x = 4cos3 t, |
y = sin3 t, |
0 ≤ t ≤ π . |
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7.1.15. |
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∫ x2 dl |
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where L is the arc of the curve y = ln x |
from |
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x1 = |
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x2 = 2 2. |
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3 to the point |
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7.1.16. |
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∫ xy2 dl |
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where L is the quarter of the circumference x2 + y2 |
= 4, |
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x ≥ 0, |
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y ≤ 0. |
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173 |
7.1.17. |
∫ |
x2 + y2 dl |
where |
L is |
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curve |
x = cos t + t sin t, |
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0 ≤ t ≤ 2π . |
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y = sin t − t cost, |
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7.1.18. |
∫ |
x2 + y2 dl |
where |
L is |
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of |
the |
circumference |
x = cos t, |
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π . |
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y = sin t, 0 ≤ t ≤ |
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7.1.19. ∫ x2dl |
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where L is the arc of the curve |
x = cost + t sint, |
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y = sint − t cost, |
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0 ≤ t ≤ π . |
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7.1.20. ∫ ydl |
where L is the arc of the previous variant. |
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7.1.21. ∫ |
x2 + y2 dl |
where L is a quarter of the circumference |
x2 + y2 = 9, |
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x ≥ 0, y ≥ |
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0. |
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7.1.22. |
∫ x2 dl |
where L is a |
quarter |
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the |
circumference |
x2 + y2 = 25, |
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x ≤ 0, y ≥ |
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0. |
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7.1.23. ∫ (x2 + yz)dl |
if L is a line segment connecting two points А(1; –2; 3) |
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and В(5; 0; 2). |
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where L is the arc of the curve x = 3(t − sin t), |
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7.1.24. ∫ xdl |
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0 ≤ t ≤ 2π. . |
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y = 3(1− cos t), |
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x = 3cos3 t, |
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7.1.25. ∫ xdl |
where L is the arc of the curve y = 3sin3 t, |
0 |
≤ t |
≤ |
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7.1.26. ∫ − ydl |
where L is the arc of the curve x = cos3 t, |
y = sin3 t, |
0 ≤ t ≤ π . |
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7.1.27. |
∫ xydl, |
where L is the arc of the screw line |
x = cos t, |
y = sin t, |
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z = t, t [0; 2π]. |
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7.1.28. |
∫ xyzdl |
where L is the arc of the screw line |
x = cos t, |
y = sin t, |
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z = 3t, 0 ≤ t ≤ 2π . |
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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,
L
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 , |
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lying under the axis Ox . |
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7.2.2. ∫ x2 ydx + x3dy where L is the parabola arc y2 |
= x, |
0 ≤ x ≤ 4, |
y ≥ 0. |
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7.2.3. |
∫ xydx + yzdy + z2 xdz |
where |
L is |
a quarter |
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the circumference |
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0 ≤ t ≤ π . |
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x = 2 cos t, |
y = 1, z = 2 sin t, |
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7.2.4. ∫ x4 ydy − y4 xdx where L is the arc of the curve |
x = |
cos t , y = sin t , |
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t [0; π / 2] . |
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7.2.5. |
∫(x2 − y2 )dy where |
L is the |
arc |
of the cubic |
parabola |
y = 2x3 , |
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0 ≤ x ≤ 1. |
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7.2.6. ∫(x2 + y2 )dx where L is the arc of the parabola |
y = 2x2 , |
2 ≤ x ≤ 4. |
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7.2.7. ∫ x2 dx + x ydy if L is a quarter of the circumference |
x2 + y2 = 25, |
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x ≥ 0, y ≥ |
0 , direction of integration is counter-clockwise. |
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7.2.8. |
∫(x − y)dx + (x + y)dy |
where L is the arc |
of the cubic parabola |
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y = 2x3 , 0 ≤ x ≤ 1. |
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7.2.9. |
∫ (x − y)dx + (x + y)dy |
if AB is a line segment connecting the points |
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AB |
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A (2; 3) and B (3; 5). |
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7.2.10. ∫(x2 − y3 )dx + (x + y)dy where |
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arc |
of |
the |
parabola |
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x = y2 − 1, 0 ≤ y ≤ 1. |
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7.2.11. |
∫ x1/ 3 ydy − y1/ 3 xdx where L is |
the |
arc of |
the |
curve |
x = cos3 t , |
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y = sin3 t , |
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t [0; π / 2] . |
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175 |
7.2.12. ∫(6xy − 1)dx + 2 y2 xdy where L is the parabola arc x = 3y2 , |
y [0; 1]. |
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7.2.13. |
∫(2x2 y − y2 )dx + 6xydy |
where L is the arc of the cubic parabola |
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y = 2x3 , 0 ≤ x ≤ 1. |
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7.2.14. ∫ 2ydx − ( y − x2 )dy |
where L is the parabola arc y = x − x2 , |
x [0; 1] . |
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7.2.15. |
∫ 2xydx + 3xy2dy |
if AВ is a segment of the straight line connecting |
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the points A (1; 1) and B (2; 4). |
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7.2.16. |
∫ (x − y2 )dx + (x + y)dy |
if AВ is a line segment connecting the |
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AB |
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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-
AB
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-
AB
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 |
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А(1; 1) and В(3; 2).
7.2.20. ∫ (2xy2 + 3x2 )dx + (2x2 y + 3y2 )dy if AВ is a line segment connecting
AB
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) |
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(x − y) |
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AB |
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and В(5; 3). |
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7.2.22. |
∫ ydx − (x2 + y)dy where L is the arc of the parabola |
y = 2x − x2 , |
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lying above the axis Ox. |
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7.2.23. |
∫ (3y2 + 4 y)dx + (6xy + 4x − 4 y)dy if AВ is a line segment connec- |
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ting the points А(0; 1) and В(2; 5). |
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7.2.24. ∫(5x − 2y)dx + x2 ydy where L is the arc y = |
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x3 , 0 ≤ x ≤ 1. |
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7.2.25. ∫(4xy − y2 )dx + 2xydy where L is the arc of the parabola |
y = 2x2 − 4x, |
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0 ≤ x ≤ 2. |
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176 |
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7.2.26. ∫3x3 ydx + xydy where L is the parabola arc y = 2x2 + 6x, 0 ≤ x ≤ 3.
L
7.2.27. ∫ 4xydx + 3xydy + zdz if AВ is a line segment connecting points А(0; 1; 5)
AB
and В(2; 1; 3). |
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7.2.28. |
∫ (2zy − 3y2 )dx + 6ydy + xdz |
if |
AВ is |
a |
line |
segment connecting |
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points А(1; –1; 0) and В(2; 3; 7). |
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7.2.29. |
∫(x2 y + 2 y2 )dx − 2xydy, |
if |
L |
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arc |
of |
the |
cubic parabola |
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y = (x − 1)3 , 0 ≤ x ≤ 1. |
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7.2.30. |
∫ (2x2 − 6z)dx + x2 ydy + (2z − 1)dz, |
if |
AВ is |
a |
segment of the |
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straight line connecting the points А(–1; 0; 4) and В(1; 4; 2). |
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7.3. Evaluate a line integral |
∫ P(x, y)dx + Q(x, y)dy over a closed contour L |
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using Green’s formula. Direction of integration is counter-clockwise. |
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7.3.1. |
∫(2x + y)dx − (3y + 2x)dy |
if |
L is a |
contour of |
the triangle whose |
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vertices are А(1; 2), В(3; 1) , С(2; 5). |
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7.3.2. |
∫ (x + y2 )dx + 4xdy |
if L is |
a contour |
of |
the |
rectangle 1 ≤ x ≤ 3 , |
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0 ≤ y ≤ 4 . |
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7.3.3. |
∫ ydx + x2dy if L is a contour formed by the parabola y = x2 and the |
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straight line. y = 2x + 3 . |
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7.3.4. ∫ y2dx + xydy if L is a contour of the rectangle −1 ≤ x ≤ 1, 0 ≤ y ≤ 3 .
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7.3.5. ∫ x2 y2 dx + xydy L is a contour formed by the parabolas y = x2 |
and y2 = x. |
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7.3.6. |
∫ y2 dx + x2 ydy if |
L is |
a |
contour of the rectangle |
−1 ≤ x ≤ 4 , |
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1 ≤ y ≤ 3. |
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7.3.7. |
∫ x3dx + (x + 2 y)dy |
if L is a contour formed by the parabola |
y = x2 |
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and the straight line y = 3x + 4. |
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7.3.8. |
∫ (2x − y)dx + (4 y + x)dy, |
if |
L is a contour of the triangle |
whose |
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vertices are А(0; 4), В(4; 0) , С(2; –2). |
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177 |
7.3.9. ∫ (x2 + y)dx + (3x − y)dy if L is a contour formed by the parabola
L
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 ,
L
1 ≤ y ≤ 4 .
7.3.11. ∫ (x − y)dx + ( y + x)dy if L is a contour of the triangle whose vertices
L
are А(0; 4), В(4; 0) , С(2; -2).
7.3.12. ∫ x2 ydx − xy2dy if L is the circumference x2 + y2 = 4.
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7.3.13. |
∫ (xy − x2 )dx + xdy if L is a contour formed by the parabola y = x2 |
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and the straight line y = x + 2 . |
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7.3.14. |
∫ ydx + (x2 + y2 )dy |
if |
L is a contour of the rectangle 0 ≤ x ≤ 4 , |
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1 ≤ y ≤ 2. |
L |
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7.3.15. ∫ (2x + y)dx + 2ydy |
if L is a contour of the triangle whose vertices |
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are А(–2; 0), В(0; 3), С(2; 1). |
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7.3.16. |
∫(x2 + 2y)dx + ydy |
if L is a contour formed by the parabolas |
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y = 2x − x2 |
and y = x2 + x − 1 . |
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7.3.17. ∫ − x2 ydx + xy2 dy if L is the circumference x2 + y2 = 9 . |
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7.3.18. |
∫ (x2 − y)dx + (x − y)dy |
if L is a contour formed by the parabola |
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y = 6x − x2 |
and the straight line |
y = 5. |
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7.3.19. |
∫ (x2 + 2y2 )dx + 2ydy if L is a contour of the rectangle −2 ≤ x ≤ 0 , |
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1 ≤ y ≤ 2 . |
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7.3.20. |
∫ ( y2 − x2 )dx + x2 dy |
if |
L is a contour of the rectangle 0 ≤ x ≤ 1, |
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−1 ≤ y ≤ 2 . |
L |
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7.3.21. ∫ y2 dx − xydy |
if L is a contour of the triangle whose vertices are А(1; 2), |
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В(3; 1), С(2; 5). |
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7.3.22. ∫ y2 dx + x2 dy |
if L is a contour of the rectangle 1 ≤ x ≤ 2 , 0 ≤ y ≤ 2 . |
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178 |
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7.3.23. ∫3ydx − (2y + x2 )dy if L is a contour formed by the parabola y = 4x − x2
L
and the straight line y = 3 .
7.3.24. ∫ xydx + x2 dy if L is the circumference x2 + y2 = 25 .
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7.3.25. ∫ (x + 4y)dx − ydy |
if L is a contour of the triangle whose vertices are |
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А(1; 2), В(3; 1), С(2; 5). |
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7.3.26. |
∫ (2x + y2 )dx + ( y − x)dy if L is a contour of the triangle |
whose |
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vertices are А(–3; 1), В(4; 1) , С(2; 3). |
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7.3.27. |
∫ (x + y)dx + x2 dy |
if L is a contour formed by the parabolas |
y = x2 |
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L |
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and y = 2 − x2 . |
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7.3.28. |
∫ (x + 3y)dx + x3dy if L is a contour of the rectangle −3 ≤ x ≤ 0 , |
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−1 ≤ y ≤ 0 . |
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7.3.29. ∫ xydx + (x + 1)dy |
if L is the circumference x2 + y2 = 16 . |
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7.3.30. ∫ (2x + y)dx + ( y − 3x)dy if L is a contour of the triangle whose vertices
L
are А(–2; 0), В(0; 3) , С(2; 1).
7.4. Verify that the given line integral is path-independent and evaluate it.
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(3; 4) |
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7.4.1. |
∫ |
( y4 + 2xy)dx + (4xy3 + x2 − 3y2 )dy. |
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(1;1) |
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(2;3) |
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7.4.2. |
∫ |
(2xy + 3y − 8x)dx + (x2 + 3x)dy . |
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(1; 0) |
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(1; 0; 4) |
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7.4.3. |
∫ |
x( y2 + z2 )dx + y(x2 + z2 )dy + z(x2 + y2 )dz. |
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(0; 0; 1) |
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(2; 4) |
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7.4.4. |
∫ |
(3x2 y4 − 4)dx + (4x3 y3 + 3y2 )dy . |
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(−4; 0) |
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(3; 2; 0) |
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7.4.5. |
∫ |
( y2 z3 + 2)dx + (2xyz3 + 1)dy + (3xy2 z2 + 2z)dz . |
(0; 0; 0)
179
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(1;3) |
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7.4.6. |
∫ |
( y3 + 2 y − 3x2 )dx + (3xy2 + 2x)dy . |
(−1; 0) |
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(3; −1) |
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7.4.7. |
∫ |
(2xy6 − 6x)dx + (6x2 y5 + 4y)dy . |
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(0; 5) |
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7.4.8. |
∫ |
(4x3 y3 + 1)dx + (3x4 y2 − 4y3 )dy . |
(−3; −1) |
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7.4.9. |
∫ |
( y + z + yz)dx + (x − z + xz)dy + (x − y + xy)dz. |
(1; 0; 0) |
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7.4.10. |
∫ |
(5x4 y3 + 9x2 )dx + (3x5 y2 + 2 y)dy . |
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(0; 0) |
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(3;3;3) |
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7.4.11. |
∫ |
(3x2 y2 z − 1)dx + (2x3 yz − 2)dy + (x3 y2 − 3)dz . |
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(0;1; 2) |
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(1;3) |
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7.4.12. |
∫ |
(3x2 y + y2 + 2x)dx + (x3 + 2xy)dy . |
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(−2; −1) |
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(1; 0; 4) |
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7.4.13. |
∫ |
( yz + 2xy2 z2 )dx + (xz + 2yx2 z2 )dy + (xy + 2zx2 y2 )dz. |
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(0; 2; 0) |
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(3;3;5) |
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7.4.14. |
∫ |
(x − yz)dx + (2 y − xz)dy + (2z − xy)dz . |
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(1; 0; 0) |
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(3; 4) |
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7.4.15. |
∫ |
(3x2 y + 2xy4 + 2x)dx + (x3 + 4x2 y3 )dy. |
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(0; −2) |
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(1; 4; 2) |
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7.4.16. |
∫ |
( yz2 + 1)dx + (xz2 + 2y)dy + (2xyz + 3z2 )dz . |
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(0; 0; 0) |
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(1; 4) |
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7.4.17. |
∫ |
(4x3 y2 + 3x2 y + 2x)dx + (2x4 y + x3 + 2y)dy . |
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(0; 0) |
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(1; 2; 3) |
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7.4.18. |
∫ |
(2xy + z2 + 3)dx + (x2 + 2 yz − 2y)dy + (2xz + y2 )dz. |
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(0; 0; 0) |
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(2; 4) |
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7.4.19. |
∫ |
(2x3 − 3y2 + 4y)dx + (4x − 6xy − 2y)dy. |
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(1;1) |
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180