Line Integrals (Exercises)
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- Jun 14, 2019
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1. True or False? Line integral ∫Cf(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 ⇀r1=tˆi+t2ˆj,0≤t≤1, and ⇀r2=(1−t)ˆi+(1−t)2ˆj,0≤t≤1, define the same oriented curve.
3. True or False? ∫−C(Pdx+Qdy)=∫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 C is given by x(t)=t,y(t)=t,0≤t≤1, then ∫Cxyds=∫10t2dt.
- Answer
- False
For the following exercises, use a computer algebra system (CAS) to evaluate the line integrals over the indicated path.
6. [T] ∫C(x+y)ds
C:x=t,y=(1−t),z=0 from (0,1,0) to (1,0,0)
7. [T] ∫C(x−y)ds
C:⇀r(t)=4tˆi+3tˆj when 0≤t≤2
- Answer
- ∫C(x−y)ds=10
8. [T] ∫C(x2+y2+z2)ds
C:⇀r(t)=sintˆi+costˆj+8tˆk when 0≤t≤π2
9. [T] Evaluate ∫Cxy4ds, where C is the right half of circle x2+y2=16 and is traversed in the clockwise direction.
- Answer
- ∫Cxy4ds=81925
10. [T] Evaluate ∫C4x3ds, 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 ⇀F(x,y,z)=xˆi+3xyˆj−(x+z)ˆk on a particle moving along a line segment that goes from (1,4,2) to (0,5,1).
- Answer
- W=8 units of work
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 ⇀F(x,y,z)=−12xˆi−12yˆj+14ˆk on a particle as it moves along the helix ⇀r(t)=costˆi+sintˆj+tˆk from point (1,0,0) to point (−1,0,3π).
- Answer
- W=3π4 units of work
14. Find the work done by vector field ⇀F(x,y)=yˆi+2xˆj in moving an object along path C, which joins points (1,0) and (0,1).
15. Find the work done by force ⇀F(x,y)=2yˆi+3xˆj+(x+y)ˆk in moving an object along curve ⇀r(t)=cos(t)ˆi+sin(t)ˆj+16ˆk, where 0≤t≤2π.
- Answer
- W=π units of work
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)=y2.
For the following exercises, evaluate the line integrals.
17. Evaluate ∫C⇀F·d⇀r, where ⇀F(x,y)=−1ˆj, and C is the part of the graph of y=12x3−x from (2,2) to (−2,−2).
- Answer
- ∫C⇀F·d⇀r=4 units of work
18. Evaluate ∫γ(x2+y2+z2)−1ds, where γ is the helix x=cost,y=sint,z=t, with 0≤t≤T.
19. Evaluate ∫Cyzdx+xzdy+xydz over the line segment from (1,1,1) to (3,2,0).
- Answer
- ∫Cyzdx+xzdy+xydz=−1
20. Let C be the line segment from point (0, 1, 1) to point (2, 2, 3). Evaluate line integral ∫Cyds.
21. [T] Use a computer algebra system to evaluate the line integral ∫Cy2dx+xdy, where C is the arc of the parabola x=4−y2 from (−5,−3) to (0,2).
- Answer
- ∫C(y2)dx+(x)dy=2456
22. [T] Use a computer algebra system to evaluate the line integral ∫C(x+3y2)dy over the path C given by x=2t,y=10t, where 0≤t≤1.
23. [T] Use a CAS to evaluate line integral ∫Cxydx+ydy over path C given by x=2t,y=10t, where 0≤t≤1.
- Answer
- ∫Cxydx+ydy=1903
24. Evaluate line integral ∫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 ∫Cy2x2−y2ds, where C is defined by the parametric equations x=t,y=t, for 1≤t≤5.
- Answer
- ∫Cy2x2−y2ds=√2ln5
27. [T] Use a CAS to evaluate ∫Cxyds, where C is defined by the parametric equations x=t2,y=4t, for 0≤t≤1.
In the following exercises, find the work done by force field ⇀F on an object moving along the indicated path.
28. ⇀F(x,y)=−xˆi−2yˆj
C:y=x3 from (0,0) to (2,8)
- Answer
- W=−66 units of work
29. ⇀F(x,y)=2xˆi+yˆj
<C: counterclockwise around the triangle with vertices (0,0),(1,0), and (1,1)
30. ⇀F(x,y,z)=xˆi+yˆj−5zˆk
C:⇀r(t)=2costˆi+2sintˆj+tˆk,0≤t≤2π
- Answer
- W=−10π2 units of work
31. Let ⇀F be vector field ⇀F(x,y)=(y2+2xey+1)ˆi+(2xy+x2ey+2y)ˆj. Compute the work of integral ∫C⇀F·d⇀r, where C is the path ⇀r(t)=sintˆi+costˆj,0≤t≤π2.
32. Compute the work done by force ⇀F(x,y,z)=2xˆi+3yˆj−zˆk along path ⇀r(t)=tˆi+t2ˆj+t3ˆk, where 0≤t≤1.
- Answer
- W=2 units of work
33. Evaluate ∫C⇀F·d⇀r, where ⇀F(x,y)=1x+yˆi+1x+yˆj and C is the segment of the unit circle going counterclockwise from (1,0) to (0,1).
34. Force ⇀F(x,y,z)=zyˆi+xˆj+z2xˆk acts on a particle that travels from the origin to point (1,2,3). Calculate the work done if the particle travels:
- along the path (0,0,0)→(1,0,0)→(1,2,0)→(1,2,3) along straight-line segments joining each pair of endpoints;
- along the straight line joining the initial and final points.
- Is the work the same along the two paths?
- Answer
- a. W=11 units of work;
b. W=394=934 units of work;
c. No
35. Find the work done by vector field ⇀F(x,y,z)=xˆi+3xyˆj−(x+z)ˆk 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 ⇀F(x,y)=yˆi+3xˆj along the upper part of ellipse x24+y2=1 from (2,0) to (−2,0)?
- Answer
- W=2π units of work
37. A vector field is given by ⇀F(x,y)=(2x+3y)ˆi+(3x+2y)ˆj. 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=x2 connecting the origin to point (1,1).
- Answer
- ∫Cfds=25√5+1120
39. Find ∫Cy2dx+(xy−x2)dy along C:y=3x from (0,0) to (1,3).
40. Find ∫Cy2dx+(xy−x2)dy along C:y2=9x from (0,0) to (1,3).
- Answer
- ∫Cy2dx+(xy−x2)dy=6.15
For the following exercises, use a CAS to evaluate the given line integrals.
41. [T] Evaluate ⇀F(x,y,z)=x2zˆi+6yˆj+yz2ˆk, where C is represented by ⇀r(t)=tˆi+t2ˆj+lntˆk,1≤t≤3.
42. [T] Evaluate line integral ∫γxeyds where, γ is the arc of curve x=ey from (1,0) to (e,1).
- Answer
- ∫γxeyds≈7.157
43. [T] Evaluate the integral ∫γxy2ds, where γ is a triangle with vertices (0,1,2),(1,0,3), and (0,−1,0).
44. [T] Evaluate line integral ∫γ(y2−xy)dx, where γ is curve y=lnx from (1,0) toward (e,1).
- Answer
- ∫γ(y2−xy)dx≈−1.379
45. [T] Evaluate line integral ∫γxy4ds, where γ is the right half of circle x2+y2=16.
46. [T] Evaluate ∫C⇀F⋅d⇀r,∫C⇀F·d⇀r, where ⇀F(x,y,z)=x2yˆi+(x−z)ˆj+xyzˆk and
C:⇀r(t)=tˆi+t2ˆj+2ˆk,0≤t≤1.
- Answer
- ∫C⇀F⋅d⇀r≈−1.133 units of work
47. Evaluate ∫C⇀F⋅d⇀r, where ⇀F(x,y)=2xsinyˆi+(x2cosy−3y2)ˆj and
C is any path from (−1,0) to (5,1).
48. Find the line integral of ⇀F(x,y,z)=12x2ˆi−5xyˆj+xzˆk over path C defined by y=x2,z=x3 from point (0,0,0) to point (2,4,8).
- Answer
- ∫C⇀F⋅d⇀r≈22.857 units of work
49. Find the line integral of ∫C(1+x2y)ds, where C is ellipse ⇀r(t)=2costˆi+3sintˆj from 0≤t≤π.
For the following exercises, find the flux.
50. Compute the flux of ⇀F=x2ˆi+yˆj across a line segment from (0,0) to (1,2).
- Answer
- flux=−13
51. Let ⇀F=5ˆi and let C be curve y=0, with 0≤x≤4. Find the flux across C.
52. Let ⇀F=5ˆj and let C be curve y=0, with 0≤x≤4. Find the flux across C.
- Answer
- flux=−20
53. Let ⇀F=−yˆi+xˆj and let C:⇀r(t)=costˆi+sintˆj for 0≤t≤2π. Calculate the flux across C.
54. Let ⇀F=(x2+y3)ˆi+(2xy)ˆj. Calculate flux ⇀F orientated counterclockwise across the curve C:x2+y2=9.
- Answer
- flux=0
Complete the rest of the exercises as stated.
55. Find the line integral of ∫Cz2dx+ydy+2ydz, where C consists of two parts: C1 and C2. C1 is the intersection of cylinder x2+y2=16 and plane z=3 from (0,4,3) to (−4,0,3). C2 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 a constant mass density of ρ grams per cm.
- Answer
- m=4πρ√5 grams
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)=y2ˆi+2(x+1)yˆj counterclockwise from point (2,0) along elliptical path x2+4y2=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 units of work
59. Find the work done when an object moves in force field ⇀F(x,y,z)=2xˆi−(x+z)ˆj+(y−x)ˆk along the path given by ⇀r(t)=t2ˆi+(t2−t)ˆj+3ˆk,0≤t≤1.
60. If an inverse force field ⇀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=\frac{k}{2} units of work
61. David and Sandra plan to evaluate line integral \displaystyle\int _C\vecs F·d\vecs{r} along a path in the xy-plane from (0, 0) to (1, 1). The force field is \vecs{F}(x,y)=(x+2y)\,\mathbf{\hat i}+(−x+y^2)\,\mathbf{\hat 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?
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.