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15.4E: Exercises for Section 15.4

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For the following exercises, evaluate the line integrals by applying Green’s theorem.

1. C2xydx+(x+y)dy, where C is the path from (0,0) to (1,1) along the graph of y=x3 and from (1,1) to (0,0) along the graph of y=x oriented in the counterclockwise direction

2. C2xydx+(x+y)dy, where C is the boundary of the region lying between the graphs of y=0 and y=4x2 oriented in the counterclockwise direction

Answer
C2xydx+(x+y)dy=323 units of work

3. C2arctan(yx)dx+ln(x2+y2)dy, where C is defined by x=4+2cosθ,y=4sinθ oriented in the counterclockwise direction

4. Csinxcosydx+(xy+cosxsiny)dy, where C is the boundary of the region lying between the graphs of y=x and y=x oriented in the counterclockwise direction

Answer
Csinxcosydx+(xy+cosxsiny)dy=112 units of work

5. Cxydx+(x+y)dy, where C is the boundary of the region lying between the graphs of x2+y2=1 and x2+y2=9 oriented in the counterclockwise direction

6. C(ydx+xdy), where C consists of line segment C1 from (1,0) to (1,0), followed by the semicircular arc C2 from (1,0) back to (1,0)

Answer
C(ydx+xdy)=π units of work

For the following exercises, use Green’s theorem.

7. Let C be the curve consisting of line segments from (0,0) to (1,1) to (0,1) and back to (0,0). Find the value of Cxydx+y2+1dy.

8. Evaluate line integral Cxe2xdx+(x4+2x2y2)dy, where C is the boundary of the region between circles x2+y2=1 and x2+y2=4, and is a positively oriented curve.

Answer
Cxe2xdx+(x4+2x2y2)dy=0 units of work

9. Find the counterclockwise circulation of field F(x,y)=xyˆi+y2ˆj around and over the boundary of the region enclosed by curves y=x2 and y=x in the first quadrant and oriented in the counterclockwise direction.

CNX_Calc_Figure_16_04_201.jpg

10. Evaluate Cy3dxx3y2dy, where C is the positively oriented circle of radius 2 centered at the origin.

Answer
Cy3dxx3y2dy=20π units of work

11. Evaluate Cy3dxx3dy, where C includes the two circles of radius 2 and radius 1 centered at the origin, both with positive orientation.

CNX_Calc_Figure_16_04_202.jpg

12. Calculate Cx2ydx+xy2dy, where C is a circle of radius 2 centered at the origin and oriented in the counterclockwise direction.

Answer
Cx2ydx+xy2dy=8π units of work

13. Calculate integral C2[y+xsin(y)]dx+[x2cos(y)3y2]dy along triangle C with vertices (0,0),(1,0) and (1,1), oriented counterclockwise, using Green’s theorem.

14. Evaluate integral C(x2+y2)dx+2xydy, where C is the curve that follows parabola y=x2 from (0,0),(2,4), then the line from (2,4) to (2,0), and finally the line from (2,0) to (0,0).

CNX_Calc_Figure_16_04_203.jpg

Answer
C(x2+y2)dx+2xydy=0 units of work

15. Evaluate line integral C(ysin(y)cos(y))dx+2xsin2(y)dy, where C is oriented in a counterclockwise path around the region bounded by x=1,x=2,y=4x2, and y=x2.

CNX_Calc_Figure_16_04_204.jpg

For the following exercises, use Green’s theorem to find the area.

16. Find the area between ellipse x29+y24=1 and circle x2+y2=25.

Answer
A=19πunits2

17. Find the area of the region enclosed by parametric equation

p(θ)=(cos(θ)cos2(θ))ˆi+(sin(θ)cos(θ)sin(θ))ˆj for 0θ2π.

CNX_Calc_Figure_16_04_205.jpg

18. Find the area of the region bounded by hypocycloid r(t)=cos3(t)ˆi+sin3(t)ˆj. The curve is parameterized by t[0,2π].

Answer
A=38πunits2

19. Find the area of a pentagon with vertices (0,4),(4,1),(3,0),(1,1), and (2,2).

20. Use Green’s theorem to evaluate C+(y2+x3)dx+x4dy, where C+ is the perimeter of square [0,1]×[0,1] oriented counterclockwise.

Answer
C+(y2+x3)dx+x4dy=0

21. Use Green’s theorem to prove the area of a disk with radius a is A=πa2units2.

22. Use Green’s theorem to find the area of one loop of a four-leaf rose r=3sin2θ. (Hint: xdyydx=r2dθ).

Answer
A=9π8units2

23. Use Green’s theorem to find the area under one arch of the cycloid given by the parametric equations: x=tsint,y=1cost,t0.

24. Use Green’s theorem to find the area of the region enclosed by curve

r(t)=t2ˆi+(t33t)ˆj, for 3t3.

CNX_Calc_Figure_16_04_206.jpg

Answer
A=835units2

25. [T] Evaluate Green’s theorem using a computer algebra system to evaluate the integral Cxeydx+exdy, where C is the circle given by x2+y2=4 and is oriented in the counterclockwise direction.

26. Evaluate C(x2y2xy+y2)ds, where C is the boundary of the unit square 0x1,0y1, traversed counterclockwise.

Answer
C(x2y2xy+y2)ds=3

27. Evaluate C(y+2)dx+(x1)dy(x1)2+(y+2)2, where C is any simple closed curve with an interior that does not contain point (1,2) traversed counterclockwise.

28. Evaluate Cxdx+ydyx2+y2, where C is any piecewise, smooth simple closed curve enclosing the origin, traversed counterclockwise.

Answer
Cxdx+ydyx2+y2=2π

For the following exercises, use Green’s theorem to calculate the work done by force F on a particle that is moving counterclockwise around closed path C.

29. F(x,y)=xyˆi+(x+y)ˆj,C:x2+y2=4

30. F(x,y)=(x3/23y)ˆi+(6x+5y)ˆj,C: boundary of a triangle with vertices (0,0),(5,0), and (0,5)

Answer
W=2252 units of work

31. Evaluate C(2x3y3)dx+(x3+y3)dy, where C is a unit circle oriented in the counterclockwise direction.

32. A particle starts at point (2,0), moves along the x-axis to (2,0), and then travels along semicircle y=4x2 to the starting point. Use Green’s theorem to find the work done on this particle by force field F(x,y)=xˆi+(x3+3xy2)ˆj.

Answer
W=12π units of work

33. David and Sandra are skating on a frictionless pond in the wind. David skates on the inside, going along a circle of radius 2 in a counterclockwise direction. Sandra skates once around a circle of radius 3, also in the counterclockwise direction. Suppose the force of the wind at point (x,y) is F(x,y)=(x2y+10y)ˆi+(x3+2xy2)ˆj. Use Green’s theorem to determine who does more work.

34. Use Green’s theorem to find the work done by force field F(x,y)=(3y4x)ˆi+(4xy)ˆj when an object moves once counterclockwise around ellipse 4x2+y2=4.

Answer
W=2π units of work

35. Use Green’s theorem to evaluate line integral Ce2xsin2ydx+e2xcos2ydy, where C is ellipse 9(x1)2+4(y3)2=36 oriented counterclockwise.

36. Evaluate line integral Cy2dx+x2dy, where C is the boundary of a triangle with vertices (0,0),(1,1), and (1,0), with the counterclockwise orientation.

Answer
Cy2dx+x2dy=13 units of work

37. Use Green’s theorem to evaluate line integral Ch·dr if h(x,y)=eyˆisinπxˆj, where C is a triangle with vertices (1,0),(0,1), and (1,0), traversed counterclockwise.

38. Use Green’s theorem to evaluate line integral C1+x3dx+2xydy where C is a triangle with vertices (0,0),(1,0), and (1,3) oriented clockwise.

Answer
C1+x3dx+2xydy=3 units of work

39. Use Green’s theorem to evaluate line integral Cx2ydxxy2dy where C is a circle x2+y2=4 oriented counterclockwise.

40. Use Green’s theorem to evaluate line integral C(3yesinx)dx+(7x+y4+1)dy where C is circle x2+y2=9 oriented in the counterclockwise direction.

Answer
C(3yesinx)dx+(7x+y4+1)dy=36π units of work

41. Use Green’s theorem to evaluate line integral C(3x5y)dx+(x6y)dy, where C is ellipse x24+y2=1 and is oriented in the counterclockwise direction.

A horizontal oval oriented counterclockwise with vertices at (-2,0), (0,-1), (2,0), and (0,1). The region enclosed is shaded.

42. Let C be a triangular closed curve from (0,0) to (1,0) to (1,1) and finally back to (0,0). Let F(x,y)=4yˆi+6x2ˆj. Use Green’s theorem to evaluate CF·dr.

Answer
CF·dr=2 units of work

43. Use Green’s theorem to evaluate line integral Cydxxdy, where C is circle x2+y2=a2 oriented in the clockwise direction.

44. Use Green’s theorem to evaluate line integral C(y+x)dx+(x+siny)dy, where C is any smooth simple closed curve joining the origin to itself oriented in the counterclockwise direction.

Answer
C(y+x)dx+(x+siny)dy=0 units of work

45. Use Green’s theorem to evaluate line integral C(yln(x2+y2))dx+(2arctanyx)dy, where C is the positively oriented circle (x2)2+(y3)2=1.

46. Use Green’s theorem to evaluate Cxydx+x3y3dy, where C is a triangle with vertices (0,0),(1,0), and (1,2) with positive orientation.

Answer
Cxydx+x3y3dy=2221 units of work

47. Use Green’s theorem to evaluate line integral Csinydx+xcosydy, where C is ellipse x2+xy+y2=1 oriented in the counterclockwise direction.

48. Let F(x,y)=(cos(x5)13y3)ˆi+13x3ˆj. Find the counterclockwise circulation CF·dr, where C is a curve consisting of the line segment joining (2,0) and (1,0), half circle y=1x2, the line segment joining (1,0) and (2,0), and half circle y=4x2.

Answer
CF·dr=15π4 units of work

49. Use Green’s theorem to evaluate line integral Csin(x3)dx+2yex2dy, where C is a triangular closed curve that connects the points (0,0),(2,2), and (0,2) counterclockwise.

50. Let C be the boundary of square 0xπ,0yπ, traversed counterclockwise. Use Green’s theorem to find Csin(x+y)dx+cos(x+y)dy.

Answer
Csin(x+y)dx+cos(x+y)dy=4 units of work

51. Use Green’s theorem to evaluate line integral CF·dr, where F(x,y)=(y2x2)ˆi+(x2+y2)ˆj, and C is a triangle bounded by y=0,x=3, and y=x, oriented counterclockwise.

52. Use Green’s Theorem to evaluate integral CF·dr, where F(x,y)=(xy2)ˆi+xˆj, and C is a unit circle oriented in the counterclockwise direction.

Answer
CF·dr=π units of work

53. Use Green’s theorem in a plane to evaluate line integral C(xy+y2)dx+x2dy, where C is a closed curve of a region bounded by y=x and y=x2 oriented in the counterclockwise direction.

54. Calculate the outward flux of F(x,y)=xˆi+2yˆj over a square with corners (±1,±1), where the unit normal is outward pointing and oriented in the counterclockwise direction.

Answer
CF·Nds=4

55. [T] Let C be circle x2+y2=4 oriented in the counterclockwise direction. Evaluate C[(3yearctanx)dx+(7x+y4+1)dy] using a computer algebra system.

56. Find the flux of field F(x,y)=xˆi+yˆj across x2+y2=16 oriented in the counterclockwise direction.

Answer
CF·Nds=32π

57. Let F=(y2x2)ˆi+(x2+y2)ˆj, and let C be a triangle bounded by y=0,x=3, and y=x oriented in the counterclockwise direction. Find the outward flux of F through C.

58. [T] Let C be unit circle x2+y2=1 traversed once counterclockwise. Evaluate C[y3+sin(xy)+xycos(xy)]dx+[x3+x2cos(xy)]dy by using a computer algebra system.

Answer
C[y3+sin(xy)+xycos(xy)]dx+[x3+x2cos(xy)]dy=4.7124 units of work

59. [T] Find the outward flux of vector field F(x,y)=xy2ˆi+x2yˆj across the boundary of annulus R={(x,y):1x2+y24}={(r,θ):1r2,0θ2π} using a computer algebra system.

60. Consider region R bounded by parabolas y=x2 and x=y2. Let C be the boundary of R oriented counterclockwise. Use Green’s theorem to evaluate C(y+ex)dx+(2x+cos(y2))dy.

Answer
C(y+ex)dx+(2x+cos(y2))dy=13 units of work

Contributors

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

 


This page titled 15.4E: Exercises for Section 15.4 is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.

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