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5.5: Line Integrals (Exercises)

• • Contributed by OpenStax
• Mathematics at OpenStax CNX

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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 $$C: r(t)=costi+sintj (0≤t≤2π)$$. 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?