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Mathematics LibreTexts

15.2E: Line Integrals (Exercises)

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    1. True or False? Line integral \(\displaystyle\int _C f(x,y)\,ds\) is equal to a definite integral if \(C\) is a smooth curve defined on \([a,b]\) and if function \(f\) is continuous on some region that contains curve \(C\).

    Solution: True

    2. True or False? Vector functions \(\vecs r_1=t\,\hat{\mathbf i}+t^2\,\hat{\mathbf j}, \quad 0≤t≤1,\) and \(\vecs r_2=(1−t)\,\hat{\mathbf i}+(1−t)^2\,\hat{\mathbf j}, \quad 0≤t≤1\), define the same oriented curve.

    3. True or False? \(\displaystyle\int _{−C}(P\,dx+Q\,dy)=\int _C(P\,dx−Q\,dy)\)

    Solution: False

    4. True or False? A piecewise smooth curve \(C\) consists of a finite number of smooth curves that are joined together end to end.

    5. True or False? If \(C\) is given by \(x(t)=t,\quad y(t)=t, \quad 0≤t≤1\), then \(\displaystyle\int _Cxy\,ds=\int ^1_0t^2\,dt.\)

    Solution: False

    For the following exercises, use a computer algebra system (CAS) to evaluate the line integrals over the indicated path.

    6. [T] \(\displaystyle\int _C(x+y)\,ds\)

    \(C:x=t,y=(1−t),z=0\) from \((0, 1, 0)\) to \((1, 0, 0)\)

    7. [T] \(\displaystyle \int _C(x−y)ds\)

    \(C:\vecs r(t)=4t\,\hat{\mathbf i}+3t\,\hat{\mathbf j}\) when \(0≤t≤2\)

    Solution: \(\displaystyle\int _C(x−y)\,ds=10\)

    8. [T] \(\displaystyle\int _C(x^2+y^2+z^2)\,ds\)

    \(C:\vecs r(t)=sint\,\hat{\mathbf i}+cost\,\hat{\mathbf j}+8t\,\hat{\mathbf k}\) when \(0≤t≤\dfrac{π}{2}\)

    9. [T] Evaluate \(\displaystyle\int _Cxy^4\,ds\), where \(C\) is the right half of circle \(x^2+y^2=16\) and is traversed in the clockwise direction.

    Solution: \(\displaystyle\int _Cxy^4\,ds=\frac{8192}{5}\)

    10. [T] Evaluate \(\displaystyle\int _C4x^3ds\), where C is the line segment from \((−2,−1)\) to \((1, 2)\).

    For the following exercises, find the work done.

    11. Find the work done by vector field \(\vecs F(x,y,z)=x\,\hat{\mathbf i}+3xy\,\hat{\mathbf j}−(x+z)\,\hat{\mathbf k}\) on a particle moving along a line segment that goes from \((1,4,2)\) to \((0,5,1)\).

    Solution: \(W=8\)

    12. Find the work done by a person weighing 150 lb walking exactly one revolution up a circular, spiral staircase of radius 3 ft if the person rises 10 ft.

    13. Find the work done by force field \(\vecs F(x,y,z)=−\dfrac{1}{2}x\,\hat{\mathbf i}−\dfrac{1}{2}y\,\hat{\mathbf j}+\dfrac{1}{4}\,\hat{\mathbf k}\) on a particle as it moves along the helix \(\vecs r(t)=\cos t\,\hat{\mathbf i}+\sin t\,\hat{\mathbf j}+t\,\hat{\mathbf k}\) from point \((1,0,0)\) to point \((−1,0,3π)\).

    Solution: \(W=\dfrac{3π}{4}\)

    14. Find the work done by vector field \(\vecs{F}(x,y)=y\,\hat{\mathbf i}+2x\,\hat{\mathbf j}\) in moving an object along path \(C\), which joins points \((1, 0)\) and \((0, 1)\).

    15. Find the work done by force \(\vecs{F}(x,y)=2y\,\hat{\mathbf i}+3x\,\hat{\mathbf j}+(x+y)\,\hat{\mathbf k}\) in moving an object along curve \(\vecs r(t)=\cos(t)\,\hat{\mathbf i}+\sin(t)\,\hat{\mathbf j}+16\,\hat{\mathbf k}\), where \(0≤t≤2π\).

    Solution: \(W=π\)

    16. Find the mass of a wire in the shape of a circle of radius 2 centered at (3, 4) with linear mass density \(ρ(x,y)=y^2\).

    For the following exercises, evaluate the line integrals.

    17. Evaluate \(\displaystyle\int C\vecs F·d\vecs{r}\), where \(\vecs{F}(x,y)=−1\,\hat{\mathbf j}\), and \(C\) is the part of the graph of \(y=12x^3−x\) from \((2,2)\) to \((−2,−2)\).

    Solution: \(\displaystyle\int _C\vecs F·d\vecs{r}=4\)

    18. Evaluate \(\displaystyle\int _γ(x^2+y^2+z^2)^{−1}ds\), where \(γ\) is the helix \(x=\cos t,y=\sin t,z=t(0≤t≤T).\)

    19. Evaluate \(\displaystyle\int _Cyz\,dx+xz\,dy+xy\,dz\) over the line segment from \((1,1,1) \) to \((3,2,0).\)

    Solution: \(\displaystyle\int _Cyz\,dx+xz\,dy+xy\,dz=−1\)

    20. Let C be the line segment from point (0, 1, 1) to point (2, 2, 3). Evaluate line integral \(\displaystyle\int _Cy\,ds.\)

    21. [T] Use a computer algebra system to evaluate the line integral \(\displaystyle\int _Cy^2\,dx+x\,dy\), where \(C\) is the arc of the parabola \(x=4−y^2\) from \((−5, −3)\) to \((0, 2)\).

    Solution: \(\displaystyle\int _C(y^2)\,dx+(x)\,dy=\dfrac{245}{6}\) 

    22. [T] Use a computer algebra system to evaluate the line integral \(\displaystyle\int _C(x+3y^2)dy\) over the path \(C\) given by \(x=2t,y=10t,\) where \(0≤t≤1.\)

    23. [T] Use a CAS to evaluate line integral \(\displaystyle\int _Cxy\,dx+y\,dy\) over path \(C\) given by \(x=2t,y=10t\),where \(0≤t≤1\).

    Solution: \(\displaystyle\int _Cxy\,dx+y\,dy=\dfrac{190}{3}\)

    24. Evaluate line integral \(\displaystyle\int _C(2x−y)\,dx+(x+3y)\,dy\), where \(C\) lies along the \(x\)-axis from \(x=0\) to \(x=5\).

    26. [T] Use a CAS to evaluate \(\displaystyle\int _C\dfrac{y}{2x^2−y^2}\,ds\), where \(C\) is \(x=t,y=t,1≤t≤5.\)

    Solution: \(\displaystyle\int _C\dfrac{y}{2x^2−y^2}\,ds=\sqrt{2}ln5\)

    27. [T] Use a CAS to evaluate \(\displaystyle\int _Cxy\,ds\), where \(C\) is \(x=t^2,y=4t,0≤t≤1.\)

    In the following exercises, find the work done by force field \(\vecs F\) on an object moving along the indicated path.

    28. \(\vecs{F}(x,y)=−x \,\hat{\mathbf i}−2y\,\hat{\mathbf j}\)

    \(C:y=x^3\) from \((0, 0)\) to \((2, 8)\)

    Solution: \(W=−66\)

    29. \(\vecs{F}(x,y)=2x\,\hat{\mathbf i}+y\,\hat{\mathbf j}\)

    C: counterclockwise around the triangle with vertices \((0, 0), (1, 0), \) and \((1, 1)\)

    30. \(\vecs F(x,y,z)=x\,\hat{\mathbf i}+y\,\hat{\mathbf j}−5z\,\hat{\mathbf k}\)

    \(C:\vecs r(t)=2\cos t\,\hat{\mathbf i}+2\sin t\,\hat{\mathbf j}+t\,\hat{\mathbf k},0≤t≤2π\)

    Solution: \(W=−10π^2\)

    31. Let \(\vecs F\) be vector field \(\vecs{F}(x,y)=(y^2+2xe^y+1)\,\hat{\mathbf i}+(2xy+x^2e^y+2y)\,\hat{\mathbf j}\). Compute the work of integral \(\displaystyle\int _C\vecs F·d\vecs{r}\), where \(C\) is the path \(\vecs r(t)=\sin t\,\hat{\mathbf i}+\cos t\,\hat{\mathbf j},\quad 0≤t≤\dfrac{π}{2}\).

    32. Compute the work done by force \(\vecs F(x,y,z)=2x\,\hat{\mathbf i}+3y\,\hat{\mathbf j}−z\,\hat{\mathbf k}\) along path \(\vecs r(t)=t\,\hat{\mathbf i}+t^2\,\hat{\mathbf j}+t^3\,\hat{\mathbf k}\),where \(0≤t≤1\).

    Solution: \(W=2\)

    33. Evaluate \(\displaystyle\int _C\vecs F·d\vecs{r}\), where \(\vecs{F}(x,y)=\dfrac{1}{x+y}\,\hat{\mathbf i}+\dfrac{1}{x+y}\,\hat{\mathbf j}\) and \(C\) is the segment of the unit circle going counterclockwise from \((1,0)\) to \((0, 1)\).

    34. Force \(\vecs F(x,y,z)=zy\,\hat{\mathbf i}+x\,\hat{\mathbf j}+z^2x\,\hat{\mathbf k}\) acts on a particle that travels from the origin to point (1, 2, 3). Calculate the work done if the particle travels:

    1. along the path \((0,0,0)→(1,0,0)→(1,2,0)→(1,2,3)\) along straight-line segments joining each pair of endpoints;
    2. along the straight line joining the initial and final points.
    3. Is the work the same along the two paths?


    Solution: a. \(W=11\); b. \(W=11\); c. Yes

    35. Find the work done by vector field \(\vecs F(x,y,z)=x\,\hat{\mathbf i}+3xy\,\hat{\mathbf j}−(x+z)\,\hat{\mathbf k}\) on a particle moving along a line segment that goes from \((1, 4, 2)\) to \((0, 5, 1).\)

    36. How much work is required to move an object in vector field \(\vecs{F}(x,y)=y\,\hat{\mathbf i}+3x\,\hat{\mathbf j}\) along the upper part of ellipse \(\dfrac{x^2}{4}+y^2=1\) from \((2, 0)\) to \((−2,0)\)?

    Solution: \(W=2π\)

    37. A vector field is given by \(\vecs{F}(x,y)=(2x+3y)\,\hat{\mathbf i}+(3x+2y)\,\hat{\mathbf j}\). Evaluate the line integral of the field around a circle of unit radius traversed in a clockwise fashion.

    38. Evaluate the line integral of scalar function xy along parabolic path \(y=x^2\) connecting the origin to point \((1, 1)\).

    Solution: \(\displaystyle\int _C\vecs F·d\vecs{r}=\dfrac{25\sqrt{5}+1}{120}\)

    39. Find \(\displaystyle\int _Cy^2\,dx+(xy−x^2)\,dy\) along \(C: y=3x\) from (0, 0) to (1, 3).

    40. Find \(\displaystyle\int _Cy^2\,dx+(xy−x^2)\,dy\) along \(C: y^2=9x\) from (0, 0) to (1, 3).

    Solution: \(\displaystyle\int _Cy^2\,dx+(xy−x^2)\,dy=6.15\)

    For the following exercises, use a CAS to evaluate the given line integrals.

    41. [T] Evaluate \(\vecs F(x,y,z)=x^2z\,\hat{\mathbf i}+6y\,\hat{\mathbf j}+yz^2\,\hat{\mathbf k}\), where \(C\) is represented by \(\vecs r(t)=t\,\hat{\mathbf i}+t^2\,\hat{\mathbf j}+\ln t \,\hat{\mathbf k},1≤t≤3\).

    42. [T] Evaluate line integral \(\displaystyle\int _γxe^y\,ds\) where, \(γ\) is the arc of curve \(x=e^y\) from \((1,0)\) to \((e,1)\).

    Solution: \(\displaystyle\int _γxe^y\,ds≈7.157\)

    43. [T] Evaluate the integral \(\displaystyle\int _γxy^2\,ds\), where \(γ\) is a triangle with vertices \((0, 1, 2), (1, 0, 3)\), and \((0,−1,0)\).

    44. [T] Evaluate line integral \(\displaystyle\int _γ(y^2−xy)\,dx\), where \(γ\) is curve \(y=\ln x\) from \((1, 0)\) toward \((e,1)\).

    Solution: \(\displaystyle\int _γ(y^2−xy)\,dx≈−1.379\)

    45. [T] Evaluate line integral \(\displaystyle\int γxy4\,ds\), where \(γ\) is the right half of circle \(x^2+y^2=16\).

    46. [T] Evaluate \int CF⋅dr,\int CF·dr, where F(x,y,z)=x2yi+(x−z)j+xyzkF(x,y,z)=x2yi+(x−z)j+xyzk and

    \(C: r(t)=ti+t^2j+2k,0≤t≤1\).

    Solution: \(\displaystyle\int _CF⋅dr≈−1.133\)

    47. Evaluate \(\displaystyle\int _CF⋅dr\), where \(\vecs{F}(x,y)=2xsin(y)i+(x^2cos(y)−3y^2)j\) and

    \(C\) is any path from \((−1,0)\) to \((5, 1)\).

    48. Find the line integral of \(F(x,y,z)=12x^2i−5xyj+xzk\) over path \(C\) defined by \(y=x^2, z=x^3\) from point \((0, 0, 0)\) to point \((2, 4, 8)\).

    Solution: \(\displaystyle\int _CF⋅dr≈22.857\)

    49. Find the line integral of \(\displaystyle\int _C(1+x^2y)ds\), where \(C\) is ellipse \(r(t)=2costi+3sintj\) from \(0≤t≤π.\)\

    For the following exercises, find the flux.

    50. Compute the flux of \(\vecs{F}=x^2i+yj\) across a line segment from \((0, 0)\) to \((1, 2).\)

    Solution: \(flux=−\dfrac{1}{3}\)

    51. Let \(\vecs{F}=5i\) and let \(C\) be curve \(y=0,0≤x≤4\). Find the flux across \(C\).

    52. Let \(\vecs{F}=5j\) and let \(C\) be curve \(y=0,0≤x≤4\). Find the flux across \(C\).

    Solution: \(flux=−20\)

    53. Let \(\vecs{F}=−yi+xj\) and let \(Cr(t)=costi+sintj (0≤t≤2π)\). Calculate the flux across \(C\).

    54. Let \(\vecs{F}=(x^2+y^3)i+(2xy)j\). Calculate flux F orientated counterclockwise across curve \(C: x^2+y^2=9.\)

    Solution: \(flux=0\)

    55. Find the line integral of \(\displaystyle\int _Cz^2dx+ydy+2ydz,\) where \(C\) consists of two parts: \(C_1\) and \(C_2. C_1\) is the intersection of cylinder \(x^2+y^2=16\) and plane \(z=3\) from \((0, 4, 3)\) to \((−4,0,3). C_2\) is a line segment from \((−4,0,3)\) to \((0, 1, 5)\).

    56. A spring is made of a thin wire twisted into the shape of a circular helix \(x=2cost,y=2sint,z=t.\) Find the mass of two turns of the spring if the wire has constant mass density.

    Solution: \(m=4πρ\sqrt{5}\)

    57. A thin wire is bent into the shape of a semicircle of radius a. If the linear mass density at point P is directly proportional to its distance from the line through the endpoints, find the mass of the wire.

    58. An object moves in force field \(F(x,y,z)=y^2i+2(x+1)yj\) counterclockwise from point \((2, 0)\) along elliptical path \(x^2+4y^2=4\) to \((−2,0)\), and back to point \((2, 0)\) along the x-axis. How much work is done by the force field on the object?

    Solution: \(W=0\)

    59. Find the work done when an object moves in force field \(F(x,y,z)=2xi−(x+z)j+(y−x)k\) along the path given by \(r(t)=t^2i+(t^2−t)j+3k, 0≤t≤1.\)

    60. If an inverse force field F is given by \(F(x,y,z)=\dfrac{k}{‖r‖^3}r\), where k is a constant, find the work done by F as its point of application moves along the x-axis from \(A(1,0,0)\) to \(B(2,0,0)\).

    Solution: \(W=\dfrac{k}{2}\)

    61. David and Sandra plan to evaluate line integral \(\displaystyle\int _CF·dr\) along a path in the xy-plane from (0, 0) to (1, 1). The force field is \(\vecs{F}(x,y)=(x+2y)i+(−x+y2)j\). David chooses the path that runs along the x-axis from (0, 0) to (1, 0) and then runs along the vertical line x=1 from (1, 0) to the final point (1, 1). Sandra chooses the direct path along the diagonal line y=x from (0, 0) to (1, 1). Whose line integral is larger and by how much?