
# 15.4E: Exercises for Section 15.4


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

1. $$\displaystyle \int_C 2xy\,dx+(x+y)\,dy$$, where $$C$$ is the path from $$(0, 0)$$ to $$(1, 1)$$ along the graph of $$y=x^3$$ and from $$(1, 1)$$ to $$(0, 0)$$ along the graph of $$y=x$$ oriented in the counterclockwise direction

2. $$\displaystyle \int_C 2xy\,dx+(x+y)\,dy$$, where $$C$$ is the boundary of the region lying between the graphs of $$y=0$$ and $$y=4−x^2$$ oriented in the counterclockwise direction

$$\displaystyle \int_C2xy\,dx+(x+y)\,dy=\frac{32}{3}$$ units of work

3. $$\displaystyle \int_C 2\arctan\left(\frac{y}{x}\right)\,dx+\ln(x^2+y^2)\,dy$$, 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

$$\displaystyle \int_C\sin x\cos y\,dx+(xy+\cos x\sin y)\,dy=\frac{1}{12}$$ 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 $$(1, 0)$$

$$\displaystyle ∮_C (−y\,\,dx+x\,\,dy)=π$$ 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.

$$\displaystyle \int_C xe^{−2x}\,dx+(x^4+2x^2y^2)\,dy=0$$ 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.

$$\displaystyle ∮_C y^3\,dx−x^3y^2\,dy=−24π$$ 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.

$$\displaystyle ∮_C −x^2y\,dx+xy^2\,dy=8π$$ 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 $$(0, 0)$$.

$$\displaystyle ∮_C (x^2+y^2)\,dx+2xy\,dy=0$$ 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 $$x^2+y^2=25$$.

$$A=19π\;\text{units}^2$$

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 $$t∈[0,2π].$$

$$A=\frac{3}{8π}\;\text{units}^2$$

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.

$$\displaystyle \int_{C^+} (y^2+x^3)\,dx+x^4\,dy=0$$

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: $$x\,dy−y\,dx=r^2\,dθ$$).

$$A=\frac{9π}{8}\;\text{units}^2$$

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 $$−\sqrt{3}≤t≤\sqrt{3}$$.

$$A=\frac{8\sqrt{3}}{5}\;\text{units}^2$$

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.

$$\displaystyle \int_C (x^2y−2xy+y^2)\,ds=3$$

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.

$$\displaystyle \int_C \frac{x\,dx+y\,dy}{x^2+y^2}=2π$$

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 $$(0, 5)$$

$$W=\frac{225}{2}$$ 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 $$\vecs F(x,y)=x\,\mathbf{\hat i}+(x^3+3xy^2)\,\mathbf{\hat j}$$.

$$W=12π$$ 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 $$4x^2+y^2=4.$$

$$W=2π$$ 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.

$$\displaystyle ∮_C y^2\,dx+x^2\,dy=\frac{1}{3}$$ 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.

$$\displaystyle \int_C \sqrt{1+x^3}\,dx+2xy\,dy=3$$ 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.

$$\displaystyle \int_C \left(3y−e^{\sin x}\right)\,dx+\left(7x+\sqrt{y^4+1}\right)\,dy=36π$$ 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 $$\displaystyle ∮_C\vecs F·d\vecs r.$$

$$\displaystyle ∮_C\vecs F·d\vecs r=2$$ 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.

$$\displaystyle ∮_C (y+x)\,dx+(x+\sin y)\,dy=0$$ 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.

$$\displaystyle ∮_C xy\,dx+x^3y^3\,dy=2221$$ 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 $$y=\sqrt{4−x^2}.$$

$$\displaystyle ∮_C\vecs F·d\vecs r=15π^4$$ 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 $$\displaystyle ∫_C \sin(x+y)\,dx+\cos(x+y)\,dy.$$

$$\displaystyle \int_C\sin(x+y)\,dx+\cos(x+y)\,dy=4$$ 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.

$$\displaystyle ∫_C \vecs F·d\vecs r=π$$ 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.

$$\displaystyle ∮_C\vecs F·\vecs N \,ds=4$$

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.

$$\displaystyle ∮_C \vecs F·\vecs N\,ds=32π$$

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.

$$\displaystyle ∫_C \left[−y^3+\sin(xy)+xy\cos(xy)\right]\,dx+\left[x^3+x^2\cos(xy)\right]\,dy=4.7124$$ 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 $$\displaystyle ∮_C \left(y+e^{\sqrt{x}}\right)\,dx+\left(2x+\cos(y^2)\right)\,dy.$$

$$\displaystyle ∮_C \left(y+e^{\sqrt{x}}\right)\,dx+\left(2x+\cos(y^2)\right)\,dy=13$$ units of work