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9.2E: Exercises

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    26273
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    Exercise \(\PageIndex{1}\)

    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 \( C \).

    Answer

    True

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

    3. True or False? \(\displaystyle \int _{−C}(Pdx+Qdy)=\displaystyle \int _C(Pdx−Qdy)\)

    Answer

    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 is given by \(x(t)=t,y(t)=t, 0≤t≤1\), then \(\displaystyle \int _Cxyds=\displaystyle \int ^1_0t^2dt.\)

    Answer

    False

    Exercise \(\PageIndex{2}\)

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

    6. [T] \(\displaystyle \int _C(x+y)ds\), where \(C:x=t,y=(1−t),z=0\) from \((0, 1, 0)\) to \((1, 0, 0)\)

    7. [T] \(\displaystyle \int _C(x−y)ds\), where \( C: \vecs{r}(t)= 4t \, \hat{\mathbf i} + 3 t \, \hat{\mathbf j} , 0 \leq t\leq 2.\)

    Answer

    \(10\)

    8. [T] \(\displaystyle \int _C(x^2+y^2+z^2)ds\) where \( C: \vecs{r}(t)= \sin(t) \, \hat{\mathbf i} + \cos(t) \, \hat{ \mathbf j}+ 8t \, \hat{\mathbf k} , 0 \leq t\leq \dfrac{π}{2}.\)

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

    Answer

    \(\frac{8192}{5}\)

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

    Exercise \(\PageIndex{3}\)

    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 \((0,5,1)\).

    Answer

    \(8\)

    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 \((−1,0,3π)\).

    Answer

    \(W=\dfrac{3π}{4}\)

    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 \(0≤t≤2π\).

    Answer

    \(W=π\)

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

    Exercise \(\PageIndex{4}\)

    For the following exercises, evaluate the line integrals.

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

    Answer

    \(4\)

    18. Evaluate \(\displaystyle \int _C(x^2+y^2+z^2)^{−1}ds\), where \(C\) is the helix \(x=cost,y=sint,z=t, (0≤t≤T).\)

    19. Evaluate \(\displaystyle \int _Cyzdx+xzdy+xydz\) over the line segment from \((1,1,1) \) to \((3,2,0).\)

    Answer

    \(\displaystyle \int _Cyzdx+xzdy+xydz=−1\)

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

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

    Answer

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

    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 \(\displaystyle \int _Cxydx+ydy\) over path \( C \) given by \(x=2t,y=10t\),where \(0≤t≤1\).

    Answer

    \(\displaystyle \int _Cxydx+ydy=\dfrac{190}{3}\)

    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 \(x=t,y=t,1≤t≤5.\)

    Answer

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

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

    Exercise \(\PageIndex{5}\)

    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 \((2, 8)\)

    Answer

    \(W=−66\)

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

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

    Answer

    \(W=−10π^2\)

    Exercise \(\PageIndex{6}\)

    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 \(\displaystyle \int _C\vecs{F}·\vecs{dr}\), where \( C \) is the path \(\vecs{r}(t)=sint \, \hat{\mathbf i}+cost \, \hat{\mathbf j}, 0≤t≤\dfrac{π}{2}\).

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

    Answer

    \(W=2\)

    33. Evaluate \(\displaystyle \int _C \vecs{F}·\vecs{dr}\), 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?

      A curve and vector field in three dimensions. The curve segments go from (1,2,3) to (0,0,0) to (1,0,0) to (1,2,0), and the arrowheads point to (0,0,0), (1,0,0), and (1,2,0). The surrounding vectors are larger the more the z component increases.

    Answer

    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 \(\vecs{F}(x,y)=yi+3x \, \hat{\mathbf j}\) along the upper part of ellipse \(\dfrac{x^2}{4}+y^2=1\) from \((2, 0)\) to \((−2,0)\)?

    Answer

    \(W=2π\)

    37. A vector field is given by \(\vecs{F}(x,y)=(2x+3y)i+(3x+2y) \, \hat{\mathbf j}\). 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 \((1, 1)\).

    Answer

    \(\displaystyle \int _C \vecs{F}·\vecs{dr}=\dfrac{25\sqrt{5}+1}{120}\)

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

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

    Answer

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

    Exercise \(\PageIndex{7}\)

    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 \(\vecs{r}(t)=t \, \hat{\mathbf i}+t^2 \, \hat{\mathbf j}+lnt \, \hat{\mathbf k},1≤t≤3\).

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

    Answer

    \(\displaystyle \int _Cxe^yds≈7.157\)

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

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

    Answer

    \(\displaystyle \int _C(y^2−xy)dx≈−1.379\)

    45. [T] Evaluate line integral \(\displaystyle \int_C xy^4ds,\) where \(C\) is the right half of circle \(x^2+y^2=16.\)

    46. [T] Evaluate \(\displaystyle \int_C\vecs{F}⋅\vecs{dr}\), where \(\vecs{F}(x,y,z)=x^2y \, \hat{\mathbf i}+(x−z) \, \hat{\mathbf j}+xyz \, \hat{\mathbf k}\) and

    \(C: \vecs{r}(t)=t \, \hat{\mathbf i}+t^2 \, \hat{\mathbf j}+2 \, \hat{\mathbf k}, 0≤t≤1\).

    Answer

    \(\displaystyle \int _C\vecs{F}⋅\vecs{dr}≈−1.133\)

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

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

    48. Find the line integral of \(\vecs{F}(x,y,z)=12x^2 \, \hat{\mathbf i}−5xy \, \hat{\mathbf j}+xz \, \hat{\mathbf k}\) over path \(C\) defined by \(y=x^2, z=x^3\) from point \((0, 0, 0)\) to point \((2, 4, 8)\).

    Answer

    \(\displaystyle \int _C\vecs{F}⋅\vecs{dr}≈22.857\)

    49. Find the line integral of \(\displaystyle \int _C(1+x^2y)ds\), where \(C\) is ellipse \(\vecs{r}(t)=2cost \, \hat{\mathbf i}+3sint \, \hat{\mathbf j}\) from \(0≤t≤π.\)

    Answer

    TBA

    Exercise \(\PageIndex{8}\)

    For the following exercises, find the flux.

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

    Answer

    \(Flux=−\dfrac{1}{3}\)

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

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

    Answer

    \(Flux=−20\)

    53. Let \(\vecs{F}=−y \, \hat{\mathbf i}+x \, \hat{\mathbf j}\) and let \(C: \vecs{r}(t)=cost \, \hat{\mathbf i}+sint \, \hat{\mathbf j} (0≤t≤2π)\). Calculate the flux across \(C\).

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

    Answer

    \(Flux=0\)

    Exercise \(\PageIndex{9}\)

    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.

    Answer

    \(m=4πρ\sqrt{5}\)

    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 \(\vecs{F}(x,y,z)=y^2 \, \hat{\mathbf i}+2(x+1)y \, \hat{\mathbf j}\) 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?

    Answer

    \(W=0\)

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

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

    Answer

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

    61. David and Sandra plan to evaluate line integral \(\displaystyle \int _C \vecs{F}·\vecs{dr}\) along a path in the xy-plane from (0, 0) to (1, 1). The force field is \(\vecs{F}(x,y)=(x+2y) \, \hat{\mathbf i}+(−x+y^2) \, \hat{\mathbf j}\). 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?

    Answer

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    This page titled 9.2E: Exercises is shared under a not declared license and was authored, remixed, and/or curated by Pamini Thangarajah.

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