James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 16.2 Line Integrals 1085
17. Let F be the vector field shown in the figure.
(a) If C 1 is the vertical line segment from s23, 23d to
s23, 3d, determine whether y C1
F dr is positive, negative,
or zero.
(b) If C 2 is the counterclockwise-oriented circle with
radius 3 and center the origin, determine whether
y C2
F dr is positive, negative, or zero.
24. y C
F dr, where Fsx, y, zd − yze x i 1 zxe y j 1 xye z k and
rstd − sin t i 1 cos t j 1 tan t k, 0 < t < y4
25. y C
xy arctan z ds, where C has parametric equations
x − t 2 , y − t 3 , z − st , 1 < t < 2
26. y C
z lnsx 1 yd ds, where C has parametric equations
x − 1 1 3t, y − 2 1 t 2 , z − t 4 , 21 < t < 1
y
3
2
CAS
27–28 Use a graph of the vector field F and the curve C to
guess whether the line integral of F over C is positive, negative,
or zero. Then evaluate the line integral.
1
_3 _2 _1
0
_1
_2
_3
1
2 3
x
27. Fsx, yd − sx 2 yd i 1 xy j,
C is the arc of the circle x 2 1 y 2 − 4 traversed counterclockwise
from (2, 0) to s0, 22d
x
y
28. Fsx, yd − i 1 j,
sx 2 1 y 2 sx 2 1 y 2
C is the parabola y − 1 1 x 2 from s21, 2d to (1, 2)
18. The figure shows a vector field F and two curves C 1 and C 2.
Are the line integrals of F over C 1 and C 2 positive, negative,
or zero? Explain.
C¡
19–22 Evaluate the line integral y C
F dr, where C is given by
the vector function rstd.
19. Fsx, yd − xy 2 i 2 x 2 j,
rstd − t 3 i 1 t 2 j, 0 < t < 1
20. Fsx, y, zd − sx 1 y 2 d i 1 xz j 1 sy 1 zd k,
rstd − t 2 i 1 t 3 j 2 2t k, 0 < t < 2
y
C
21. Fsx, y, zd − sin x i 1 cos y j 1 xz k,
rstd − t 3 i 2 t 2 j 1 t k, 0 < t < 1
22. Fsx, y, zd − x i 1 y j 1 xy k,
rstd − cos t i 1 sin t j 1 t k, 0 < t <
23–26 Use a calculator to evaluate the line integral correct to
four decimal places.
23. y C
F dr, where Fsx, yd − sx 1 y i 1 syyxd j and
rstd − sin 2 t i 1 sin t cos t j, y6 < t < y3
x
;
;
CAS
CAS
29. (a) Evaluate the line integral y C
F dr, where
Fsx, yd − e x21 i 1 xy j and C is given by
rstd − t 2 i 1 t 3 j, 0 < t < 1.
(b) Illustrate part (a) by using a graphing calculator or computer
to graph C and the vectors from the vector field
corresponding to t − 0, 1ys2 , and 1 (as in Figure 13).
30. (a) Evaluate the line integral y C
F dr, where
Fsx, y, zd − x i 2 z j 1 y k and C is given by
rstd − 2t i 1 3t j 2 t 2 k, 21 < t < 1.
(b) Illustrate part (a) by using a computer to graph C and
the vectors from the vector field corresponding to
t − 61 and 6 1 2 (as in Figure 13).
31. Find the exact value of y C
x 3 y 2 z ds, where C is the curve
with parametric equations x − e 2t cos 4t, y − e 2t sin 4t,
z − e 2t , 0 < t < 2.
32. (a) Find the work done by the force field
Fsx, yd − x 2 i 1 xy j on a particle that moves once
around the circle x 2 1 y 2 − 4 oriented in the counterclockwise
direction.
(b) Use a computer algebra system to graph the force field
and circle on the same screen. Use the graph to explain
your answer to part (a).
33. A thin wire is bent into the shape of a semicircle
x 2 1 y 2 − 4, x > 0. If the linear density is a constant k,
find the mass and center of mass of the wire.
34. A thin wire has the shape of the first-quadrant part of the
circle with center the origin and radius a. If the density
function is sx, yd − kxy, find the mass and center of mass
of the wire.
35. (a) Write the formulas similar to Equations 4 for the center
of mass sx, y, zd of a thin wire in the shape of a space
curve C if the wire has density function sx, y, zd.
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