# 15.4E: Green's Theorem (Exercises)

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**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

**Answer:**- \(\displaystyle \int_C2xy\,dx+(x+y)\,dy=\frac{32}{3}\) 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

**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:**- \(\displaystyle ∮_C y^3\,dx−x^3y^2\,dy=−24π\) 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.

**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.