5.5E: Green's Theorem (Exercises)
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- Dec 18, 2020
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( \newcommand{\kernel}{\mathrm{null}\,}\)
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=4−x2 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+2\cos θ,\;y=4\sin θ oriented in the counterclockwise direction
4. \displaystyle \int_C \sin x\cos y\,dx+(xy+\cos x\sin y)\,dy, where C is the boundary of the region lying between the graphs of y=x and y=\sqrt{x} oriented in the counterclockwise direction
- Answer
- \displaystyle \int_C\sin x\cos y\,dx+(xy+\cos x\sin y)\,dy=\frac{1}{12} units of work
5. \displaystyle \int_C xy\,dx+(x+y)\,dy, where C is the boundary of the region lying between the graphs of x^2+y^2=1 and x^2+y^2=9 oriented in the counterclockwise direction
6. \displaystyle ∮_C (−y\,dx+x\,dy), where C consists of line segment C_1 from (−1,0) to (1, 0), followed by the semicircular arc C_2 from (1, 0) back to (-1, 0)
- Answer
- \displaystyle ∮_C (−y\,\,dx+x\,\,dy)=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 \displaystyle \int_C xy\,dx+\sqrt{y^2+1}\,dy.
8. Evaluate line integral \displaystyle \int_C xe^{−2x}\,dx+(x^4+2x^2y^2)\,dy, where C is the boundary of the region between circles x^2+y^2=1 and x^2+y^2=4, and is a positively oriented curve.
- Answer
- \displaystyle \int_C xe^{−2x}\,dx+(x^4+2x^2y^2)\,dy=0 units of work
9. Find the counterclockwise circulation of field \vecs F(x,y)=xy\,\mathbf{\hat i}+y^2\,\mathbf{\hat j} around and over the boundary of the region enclosed by curves y=x^2 and y=x in the first quadrant and oriented in the counterclockwise direction.
10. Evaluate \displaystyle ∮_C y^3\,dx−x^3y^2\,dy, where C is the positively oriented circle of radius 2 centered at the origin.
- Answer
- \displaystyle ∮_C y^3\,dx−x^3y^2\,dy=−20π units of work
11. Evaluate \displaystyle ∮_C y^3\,dx−x^3\,dy, where C includes the two circles of radius 2 and radius 1 centered at the origin, both with positive orientation.
12. Calculate \displaystyle ∮_C −x^2y\,dx+xy^2\,dy, where C is a circle of radius 2 centered at the origin and oriented in the counterclockwise direction.
- Answer
- \displaystyle ∮_C −x^2y\,dx+xy^2\,dy=8π units of work
13. Calculate integral \displaystyle ∮_C 2[y+x\sin(y)]\,dx+[x^2\cos(y)−3y^2]\,dy along triangle C with vertices (0, 0), \,(1, 0) and (1, 1), oriented counterclockwise, using Green’s theorem.
14. Evaluate integral \displaystyle ∮_C (x^2+y^2)\,dx+2xy\,dy, where C is the curve that follows parabola y=x^2 from (0,0), \,(2,4), then the line from (2, 4) to (2, 0), and finally the line from (2, 0) to (0, 0).
- Answer
- \displaystyle ∮_C (x^2+y^2)\,dx+2xy\,dy=0 units of work
15. Evaluate line integral \displaystyle ∮_C (y−\sin(y)\cos(y))\,dx+2x\sin^2(y)\,dy, where C is oriented in a counterclockwise path around the region bounded by x=−1, \,x=2, \,y=4−x^2, and y=x−2.
For the following exercises, use Green’s theorem to find the area.
16. Find the area between ellipse \frac{x^2}{9}+\frac{y^2}{4}=1 and circle x^2+y^2=25.
- Answer
- A=19π\;\text{units}^2
17. Find the area of the region enclosed by parametric equation
\vecs p(θ)=(\cos(θ)−\cos^2(θ))\,\mathbf{\hat i}+(\sin(θ)−\cos(θ)\sin(θ))\,\mathbf{\hat j} for 0≤θ≤2π.
18. Find the area of the region bounded by hypocycloid \vecs r(t)=\cos^3(t)\,\mathbf{\hat i}+\sin^3(t)\,\mathbf{\hat j}. The curve is parameterized by t∈[0,2π].
- Answer
- A=\frac{3}{8π}\;\text{units}^2
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 \displaystyle \int_{C^+}(y^2+x^3)\,dx+x^4\,dy, where C^+ is the perimeter of square [0,1]×[0,1] oriented counterclockwise.
- Answer
- \displaystyle \int_{C^+} (y^2+x^3)\,dx+x^4\,dy=0
21. Use Green’s theorem to prove the area of a disk with radius a is A=πa^2\;\text{units}^2.
22. Use Green’s theorem to find the area of one loop of a four-leaf rose r=3\sin 2θ. (Hint: x\,dy−y\,dx=r^2\,dθ).
- Answer
- A=\frac{9π}{8}\;\text{units}^2
23. Use Green’s theorem to find the area under one arch of the cycloid given by the parametric equations: x=t−\sin t,\;y=1−\cos t,\;t≥0.
24. Use Green’s theorem to find the area of the region enclosed by curve
\vecs r(t)=t^2\,\mathbf{\hat i}+\left(\frac{t^3}{3}−t\right)\,\mathbf{\hat j}, for −\sqrt{3}≤t≤\sqrt{3}.
- Answer
- A=\frac{8\sqrt{3}}{5}\;\text{units}^2
25. [T] Evaluate Green’s theorem using a computer algebra system to evaluate the integral \displaystyle \int_C xe^y\,dx+e^x\,dy, where C is the circle given by x^2+y^2=4 and is oriented in the counterclockwise direction.
26. Evaluate \displaystyle \int_C(x^2y−2xy+y^2)\,ds, where C is the boundary of the unit square 0≤x≤1,\;0≤y≤1, traversed counterclockwise.
- Answer
- \displaystyle \int_C (x^2y−2xy+y^2)\,ds=3
27. Evaluate \displaystyle \int_C \frac{−(y+2)\,dx+(x−1)\,dy}{(x−1)^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 \displaystyle \int_C \frac{x\,dx+y\,dy}{x^2+y^2}, where C is any piecewise, smooth simple closed curve enclosing the origin, traversed counterclockwise.
- Answer
- \displaystyle \int_C \frac{x\,dx+y\,dy}{x^2+y^2}=2π
For the following exercises, use Green’s theorem to calculate the work done by force \vecs F on a particle that is moving counterclockwise around closed path C.
29. \vecs F(x,y)=xy\,\mathbf{\hat i}+(x+y)\,\mathbf{\hat j}, \quad C:x^2+y^2=4
30. \vecs F(x,y)=(x^{3/2}−3y)\,\mathbf{\hat i}+(6x+5\sqrt{y})\,\mathbf{\hat j}, \quad C: boundary of a triangle with vertices (0, 0), \,(5, 0), and (0, 5)
- Answer
- W=\frac{225}{2} units of work
31. Evaluate \displaystyle \int_C (2x^3−y^3)\,dx+(x^3+y^3)\,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=\sqrt{4−x^2} to the starting point. Use Green’s theorem to find the work done on this particle by force field \vecs F(x,y)=x\,\mathbf{\hat i}+(x^3+3xy^2)\,\mathbf{\hat j}.
- 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 \vecs F(x,y)=(x^2y+10y)\,\mathbf{\hat i}+(x^3+2xy^2)\,\mathbf{\hat j}. Use Green’s theorem to determine who does more work.
34. Use Green’s theorem to find the work done by force field \vecs F(x,y)=(3y−4x)\,\mathbf{\hat i}+(4x−y)\,\mathbf{\hat j} when an object moves once counterclockwise around ellipse 4x^2+y^2=4.
- Answer
- W=2π units of work
35. Use Green’s theorem to evaluate line integral \displaystyle ∮_C e^{2x}\sin 2y\,dx+e^{2x}\cos 2y\,dy, where C is ellipse 9(x−1)^2+4(y−3)^2=36 oriented counterclockwise.
36. Evaluate line integral \displaystyle ∮_C y^2\,dx+x^2\,dy, where C is the boundary of a triangle with vertices (0,0), \,(1,1), and (1,0), with the counterclockwise orientation.
- Answer
- \displaystyle ∮_C y^2\,dx+x^2\,dy=\frac{1}{3} units of work
37. Use Green’s theorem to evaluate line integral \displaystyle \int_C \vecs h·d\vecs r if \vecs h(x,y)=e^y\,\mathbf{\hat i}−\sin πx\,\mathbf{\hat j}, 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 \displaystyle \int_C\sqrt{1+x^3}\,dx+2xy\,dy where C is a triangle with vertices (0, 0), \,(1, 0), and (1, 3) oriented clockwise.
- Answer
- \displaystyle \int_C \sqrt{1+x^3}\,dx+2xy\,dy=3 units of work
39. Use Green’s theorem to evaluate line integral \displaystyle \int_C x^2y\,dx−xy^2\,dy where C is a circle x^2+y^2=4 oriented counterclockwise.
40. Use Green’s theorem to evaluate line integral \displaystyle \int_C \left(3y−e^{\sin x}\right)\,dx+\left(7x+\sqrt{y^4+1}\right)\,dy where C is circle x^2+y^2=9 oriented in the counterclockwise direction.
- Answer
- \displaystyle \int_C \left(3y−e^{\sin x}\right)\,dx+\left(7x+\sqrt{y^4+1}\right)\,dy=36π units of work
41. Use Green’s theorem to evaluate line integral \displaystyle \int_C (3x−5y)\,dx+(x−6y)\,dy, where C is ellipse \frac{x^2}{4}+y^2=1 and is oriented in the counterclockwise direction.

42. Let C be a triangular closed curve from (0, 0) to (1, 0) to (1, 1) and finally back to (0, 0). Let \vecs F(x,y)=4y\,\mathbf{\hat i}+6x^2\,\mathbf{\hat j}. Use Green’s theorem to evaluate \displaystyle ∮_C\vecs F·d\vecs r.
- Answer
- \displaystyle ∮_C\vecs F·d\vecs r=2 units of work
43. Use Green’s theorem to evaluate line integral \displaystyle ∮_C y\,dx−x\,dy, where C is circle x^2+y^2=a^2 oriented in the clockwise direction.
44. Use Green’s theorem to evaluate line integral \displaystyle ∮_C (y+x)\,dx+(x+\sin y)\,dy, where C is any smooth simple closed curve joining the origin to itself oriented in the counterclockwise direction.
- Answer
- \displaystyle ∮_C (y+x)\,dx+(x+\sin y)\,dy=0 units of work
45. Use Green’s theorem to evaluate line integral \displaystyle ∮_C \left(y−\ln(x^2+y^2)\right)\,dx+\left(2\arctan \frac{y}{x}\right)\,dy, where C is the positively oriented circle (x−2)^2+(y−3)^2=1.
46. Use Green’s theorem to evaluate \displaystyle ∮_C xy\,dx+x^3y^3\,dy, where C is a triangle with vertices (0, 0), \,(1, 0), and (1, 2) with positive orientation.
- Answer
- \displaystyle ∮_C xy\,dx+x^3y^3\,dy=2221 units of work
47. Use Green’s theorem to evaluate line integral \displaystyle \int_C \sin y\,dx+x\cos y\,dy, where C is ellipse x^2+xy+y^2=1 oriented in the counterclockwise direction.
48. Let \vecs F(x,y)=\left(\cos(x^5)−13y^3\right)\,\mathbf{\hat i}+13x^3\,\mathbf{\hat j}. Find the counterclockwise circulation \displaystyle ∮_C\vecs F·d\vecs r, where C is a curve consisting of the line segment joining (−2,0) and (−1,0), half circle y=\sqrt{1−x^2}, the line segment joining (1, 0) and (2, 0), and half circle y=\sqrt{4−x^2}.
- Answer
- \displaystyle ∮_C\vecs F·d\vecs r=15π^4 units of work
49. Use Green’s theorem to evaluate line integral \displaystyle ∫_C \sin(x^3)\,dx+2ye^{x^2}\,dy, 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 0≤x≤π,\;0≤y≤π, traversed counterclockwise. Use Green’s theorem to find \displaystyle ∫_C \sin(x+y)\,dx+\cos(x+y)\,dy.
- Answer
- \displaystyle \int_C\sin(x+y)\,dx+\cos(x+y)\,dy=4 units of work
51. Use Green’s theorem to evaluate line integral \displaystyle ∫_C \vecs F·d\vecs r, where \vecs F(x,y)=(y^2−x^2)\,\mathbf{\hat i}+(x^2+y^2)\,\mathbf{\hat j}, and C is a triangle bounded by y=0,\;x=3, and y=x, oriented counterclockwise.
52. Use Green’s Theorem to evaluate integral \displaystyle ∫_C \vecs F·d\vecs r, where \vecs F(x,y)=(xy^2)\,\mathbf{\hat i}+x\,\mathbf{\hat j}, and C is a unit circle oriented in the counterclockwise direction.
- Answer
- \displaystyle ∫_C \vecs F·d\vecs r=π units of work
53. Use Green’s theorem in a plane to evaluate line integral \displaystyle ∮_C (xy+y^2)\,dx+x^2\,dy, where C is a closed curve of a region bounded by y=x and y=x^2 oriented in the counterclockwise direction.
54. Calculate the outward flux of \vecs F(x,y)=−x\,\mathbf{\hat i}+2y\,\mathbf{\hat j} over a square with corners (±1,\,±1), where the unit normal is outward pointing and oriented in the counterclockwise direction.
- Answer
- \displaystyle ∮_C\vecs F·\vecs N \,ds=4
55. [T] Let C be circle x^2+y^2=4 oriented in the counterclockwise direction. Evaluate \displaystyle ∮_C \left[\left(3y−e^{\arctan x})\,dx+(7x+\sqrt{y^4+1}\right)\,dy\right] using a computer algebra system.
56. Find the flux of field \vecs F(x,y)=−x\,\mathbf{\hat i}+y\,\mathbf{\hat j} across x^2+y^2=16 oriented in the counterclockwise direction.
- Answer
- \displaystyle ∮_C \vecs F·\vecs N\,ds=32π
57. Let \vecs F=(y^2−x^2)\,\mathbf{\hat i}+(x^2+y^2)\,\mathbf{\hat j}, 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 \vecs F through C.
58. [T] Let C be unit circle x^2+y^2=1 traversed once counterclockwise. Evaluate \displaystyle ∫_C \left[−y^3+\sin(xy)+xy\cos(xy)\right]\,dx+\left[x^3+x^2\cos(xy)\right]\,dy by using a computer algebra system.
- Answer
- \displaystyle ∫_C \left[−y^3+\sin(xy)+xy\cos(xy)\right]\,dx+\left[x^3+x^2\cos(xy)\right]\,dy=4.7124 units of work
59. [T] Find the outward flux of vector field \vecs F(x,y)=xy^2\,\mathbf{\hat i}+x^2y\,\mathbf{\hat j} across the boundary of annulus R=\big\{(x,y):1≤x^2+y^2≤4\big\}=\big\{(r,θ):1≤r≤2,\,0≤θ≤2π\big\} using a computer algebra system.
60. Consider region R bounded by parabolas y=x^2 and x=y^2. Let C be the boundary of R oriented counterclockwise. Use Green’s theorem to evaluate \displaystyle ∮_C \left(y+e^{\sqrt{x}}\right)\,dx+\left(2x+\cos(y^2)\right)\,dy.
- Answer
- \displaystyle ∮_C \left(y+e^{\sqrt{x}}\right)\,dx+\left(2x+\cos(y^2)\right)\,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.