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# 16.4E: Ejercicios para la Sección 16.4

• Edwin “Jed” Herman & Gilbert Strang
• OpenStax

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Para los siguientes ejercicios, evalúe las integrales de línea aplicando el teorema de Green.

1. $$\displaystyle \int_C 2xy\,dx+(x+y)\,dy$$, donde$$C$$ es el camino de$$(0, 0)$$ a$$(1, 1)$$ lo largo de la gráfica de$$y=x^3$$ y de$$(1, 1)$$ a$$(0, 0)$$ lo largo de la gráfica de$$y=x$$ orientado en el sentido contrario a las agujas del reloj

2. $$\displaystyle \int_C 2xy\,dx+(x+y)\,dy$$, donde$$C$$ está el límite de la región que se encuentra entre las gráficas de$$y=0$$ y$$y=4−x^2$$ orientado en sentido contrario a las agujas del reloj

Contestar
$$\displaystyle \int_C2xy\,dx+(x+y)\,dy=\frac{32}{3}$$unidades de trabajo

3. $$\displaystyle \int_C 2\arctan\left(\frac{y}{x}\right)\,dx+\ln(x^2+y^2)\,dy$$, donde$$C$$ se define por$$x=4+2\cos θ,\;y=4\sin θ$$ orientado en el sentido contrario a las agujas del reloj

4. $$\displaystyle \int_C \sin x\cos y\,dx+(xy+\cos x\sin y)\,dy$$, donde$$C$$ está el límite de la región que se encuentra entre las gráficas de$$y=x$$ y$$y=\sqrt{x}$$ orientado en sentido contrario a las agujas del reloj

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

5. $$\displaystyle \int_C xy\,dx+(x+y)\,dy$$, donde$$C$$ está el límite de la región que se encuentra entre las gráficas de$$x^2+y^2=1$$ y$$x^2+y^2=9$$ orientado en sentido contrario a las agujas del reloj

6. $$\displaystyle ∮_C (−y\,dx+x\,dy)$$, donde$$C$$ consiste en un segmento$$C_1$$ de línea de$$(−1,0)$$ a$$(1, 0)$$, seguido por el arco semicircular$$C_2$$ de$$(1, 0)$$ vuelta a$$(-1, 0)$$

Contestar
$$\displaystyle ∮_C (−y\,\,dx+x\,\,dy)=π$$unidades de trabajo

Para los siguientes ejercicios, use el teorema de Green.

7. Dejar$$C$$ ser la curva que consiste en segmentos de línea de$$(0, 0)$$ a$$(1, 1)$$ a$$(0, 1)$$ y de vuelta a$$(0, 0)$$. Encuentra el valor de$$\displaystyle \int_C xy\,dx+\sqrt{y^2+1}\,dy$$.

8. Evaluar integral de línea$$\displaystyle \int_C xe^{−2x}\,dx+(x^4+2x^2y^2)\,dy$$, donde$$C$$ está el límite de la región entre círculos$$x^2+y^2=1$$ y$$x^2+y^2=4$$, y es una curva orientada positivamente.

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

9. Encuentra la circulación de campo en sentido antihorario$$\vecs F(x,y)=xy\,\mathbf{\hat i}+y^2\,\mathbf{\hat j}$$ alrededor y sobre el límite de la región encerrada por curvas$$y=x^2$$ y$$y=x$$ en el primer cuadrante y orientada en sentido antihorario.

10. Evaluar$$\displaystyle ∮_C y^3\,dx−x^3y^2\,dy$$, donde$$C$$ está el círculo de radio orientado positivamente$$2$$ centrado en el origen.

Contestar
$$\displaystyle ∮_C y^3\,dx−x^3y^2\,dy=−20π$$unidades de trabajo

11. Evaluar$$\displaystyle ∮_C y^3\,dx−x^3\,dy$$, donde$$C$$ incluye los dos círculos de radio$$2$$ y radio$$1$$ centrados en el origen, ambos con orientación positiva.

12. Calcular$$\displaystyle ∮_C −x^2y\,dx+xy^2\,dy$$, donde$$C$$ hay un círculo de radio$$2$$ centrado en el origen y orientado en sentido contrario a las agujas del reloj.

Contestar
$$\displaystyle ∮_C −x^2y\,dx+xy^2\,dy=8π$$unidades de trabajo

13. Calcular integral$$\displaystyle ∮_C 2[y+x\sin(y)]\,dx+[x^2\cos(y)−3y^2]\,dy$$ a lo largo del triángulo$$C$$ con vértices$$(0, 0), \,(1, 0)$$ y$$(1, 1)$$, orientado en sentido antihorario, usando el teorema de Green.

14. Evaluar integral$$\displaystyle ∮_C (x^2+y^2)\,dx+2xy\,dy$$, donde$$C$$ está la curva que sigue$$y=x^2$$ a la parábola desde$$(0,0), \,(2,4),$$ entonces la línea de$$(2, 4)$$ a$$(2, 0)$$, y finalmente la línea de$$(2, 0)$$ a$$(0, 0)$$.

Contestar
$$\displaystyle ∮_C (x^2+y^2)\,dx+2xy\,dy=0$$unidades de trabajo

15. Evaluar la integral de línea$$\displaystyle ∮_C (y−\sin(y)\cos(y))\,dx+2x\sin^2(y)\,dy$$, donde$$C$$ se orienta en una trayectoria en sentido antihorario alrededor de la región delimitada por$$x=−1, \,x=2, \,y=4−x^2$$, y$$y=x−2.$$

Para los siguientes ejercicios, usa el teorema de Green para encontrar el área.

16. Encuentra el área entre elipse$$\frac{x^2}{9}+\frac{y^2}{4}=1$$ y círculo$$x^2+y^2=25$$.

Contestar
$$A=19π\;\text{units}^2$$

17. Encontrar el área de la región encerrada por la ecuación paramétrica

$$\vecs p(θ)=(\cos(θ)−\cos^2(θ))\,\mathbf{\hat i}+(\sin(θ)−\cos(θ)\sin(θ))\,\mathbf{\hat j}$$para$$0≤θ≤2π.$$

18. Encontrar el área de la región delimitada por hipocicloide$$\vecs r(t)=\cos^3(t)\,\mathbf{\hat i}+\sin^3(t)\,\mathbf{\hat j}$$. La curva es parametrizada por$$t∈[0,2π].$$

Contestar
$$A=\frac{3}{8π}\;\text{units}^2$$

19. Encuentra el área de un pentágono con vértices$$(0,4), \,(4,1), \,(3,0), \,(−1,−1),$$ y$$(−2,2)$$.

20. Utilice el teorema de Green para evaluar$$\displaystyle \int_{C^+}(y^2+x^3)\,dx+x^4\,dy$$, donde$$C^+$$ está el perímetro del cuadrado$$[0,1]×[0,1]$$ orientado en sentido antihorario.

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

21. Utilice el teorema de Green para probar el área de un disco con radio$$a$$ es$$A=πa^2\;\text{units}^2$$.

22. Usa el teorema de Green para encontrar el área de un bucle de una rosa de cuatro hojas$$r=3\sin 2θ$$. (Pista:$$x\,dy−y\,dx=r^2\,dθ$$).

Contestar
$$A=\frac{9π}{8}\;\text{units}^2$$

23. Usa el teorema de Green para encontrar el área bajo un arco del cicloide dada por las ecuaciones paramétricas:$$x=t−\sin t,\;y=1−\cos t,\;t≥0.$$

24. Usa el teorema de Green para encontrar el área de la región encerrada por curva

$$\vecs r(t)=t^2\,\mathbf{\hat i}+\left(\frac{t^3}{3}−t\right)\,\mathbf{\hat j},$$para$$−\sqrt{3}≤t≤\sqrt{3}$$.

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

25. [T] Evaluar el teorema de Green utilizando un sistema de álgebra computacional para evaluar la integral$$\displaystyle \int_C xe^y\,dx+e^x\,dy$$, donde$$C$$ está el círculo dado por$$x^2+y^2=4$$ y está orientado en sentido antihorario.

26. Evaluar$$\displaystyle \int_C(x^2y−2xy+y^2)\,ds$$, donde$$C$$ está el límite de la unidad cuadrada$$0≤x≤1,\;0≤y≤1$$, atravesado en sentido antihorario.

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

27. Evaluar$$\displaystyle \int_C \frac{−(y+2)\,dx+(x−1)\,dy}{(x−1)^2+(y+2)^2}$$, donde$$C$$ está cualquier curva cerrada simple con un interior que no contenga puntos$$(1,−2)$$ atravesados en sentido antihorario.

28. Evaluar$$\displaystyle \int_C \frac{x\,dx+y\,dy}{x^2+y^2}$$, donde$$C$$ está cualquier curva cerrada simple, lisa, por tramos, que encierra el origen, atravesada en sentido antihorario.

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

Para los siguientes ejercicios, utilice el teorema de Green para calcular el trabajo realizado por la fuerza$$\vecs F$$ sobre una partícula que se mueve en sentido antihorario alrededor de un camino cerrado$$C$$.

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