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

1102 Chapter 16 Vector Calculus
CAS
4. y C
x 2 y 2 dx 1 xy dy, C consists of the arc of the parabola
y − x 2 from s0, 0d to s1, 1d and the line segments from
s1, 1d to s0, 1d and from s0, 1d to s0, 0d
5–10 Use Green’s Theorem to evaluate the line integral along
the given positively oriented curve.
5. y C
ye x dx 1 2e x dy,
C is the rectangle with vertices s0, 0d, s3, 0d, s3, 4d,
and s0, 4d
6. y C
sx 2 1 y 2 d dx 1 sx 2 2 y 2 d dy,
C is the triangle with vertices s0, 0d, s2, 1d, and s0, 1d
7. y C
(y 1 e sx ) dx 1 s2x 1 cos y 2 d dy,
C is the boundary of the region enclosed by the parabolas
y − x 2 and x − y 2
8. y C
y 4 dx 1 2xy 3 dy, C is the ellipse x 2 1 2y 2 − 2
9. y C
y 3 dx 2 x 3 dy, C is the circle x 2 1 y 2 − 4
10. y C
s1 2 y 3 d dx 1 sx 3 1 e y 2 d dy, C is the boundary of the
region between the circles x 2 1 y 2 − 4 and x 2 1 y 2 − 9
11–14 Use Green’s Theorem to evaluate y C
F dr. (Check the
orientation of the curve before applying the theorem.)
11. Fsx, yd − ky cos x 2 xy sin x, xy 1 x cos xl,
C is the triangle from s0, 0d to s0, 4d to s2, 0d to s0, 0d
12. Fsx, yd − ke 2x 1 y 2 , e 2y 1 x 2 l,
C consists of the arc of the curve y − cos x from s2y2, 0d
to sy2, 0d and the line segment from sy2, 0d to s2y2, 0d
13. Fsx, yd − ky 2 cos y, x sin yl,
C is the circle sx 2 3d 2 1 sy 1 4d 2 − 4 oriented clockwise
14. Fsx, yd − ksx 2 1 1, tan 21 xl, C is the triangle from s0, 0d
to s1, 1d to s0, 1d to s0, 0d
15–16 Verify Green’s Theorem by using a computer algebra
system to evaluate both the line integral and the double integral.
15. Psx, yd − x 3 y 4 , Qsx, yd − x 5 y 4 ,
C consists of the line segment from s2y2, 0d to sy2, 0d
followed by the arc of the curve y − cos x from sy2, 0d to
s2y2, 0d
16. Psx, yd − 2x 2 x 3 y 5 , Qsx, yd − x 3 y 8 ,
C is the ellipse 4x 2 1 y 2 − 4
17. Use Green’s Theorem to find the work done by the force
Fsx, yd − xsx 1 yd i 1 xy 2 j in moving a particle from the
origin along the x-axis to s1, 0d, then along the line segment
to s0, 1d, and then back to the origin along the y-axis.
18. A particle starts at the origin, moves along the x-axis to
s5, 0d, then along the quarter-circle x 2 1 y 2 − 25, x > 0,
y > 0 to the point s0, 5d, and then down the y-axis
back to the origin. Use Green’s Theorem to find
;
the work done on this particle by the force field
Fsx, yd − ksin x, siny 1 xy 2 1 1 3 x 3 l.
19. Use one of the formulas in (5) to find the area under one
arch of the cycloid x − t 2 sin t, y − 1 2 cos t.
20. If a circle C with radius 1 rolls along the outside of the
circle x 2 1 y 2 − 16, a fixed point P on C traces out a
curve called an epicycloid, with parametric equations
x − 5 cos t 2 cos 5t, y − 5 sin t 2 sin 5t. Graph the epicycloid
and use (5) to find the area it encloses.
21. (a) If C is the line segment connecting the point sx 1, y 1d to
the point sx 2, y 2d, show that
y x dy 2 y dx − x
C
1y 2 2 x 2 y 1
(b) If the vertices of a polygon, in counterclockwise order,
are sx 1, y 1d, sx 2, y 2d, . . . , sx n, y nd, show that the area of
the polygon is
A − 1 2 fsx1y2 2 x2y1d 1 sx2y3 2 x3y2d 1 ∙ ∙ ∙
1 sx n21y n 2 x ny n21d 1 sx ny 1 2 x 1y ndg
(c) Find the area of the pentagon with vertices s0, 0d, s2, 1d,
s1, 3d, s0, 2d, and s21, 1d.
22. Let D be a region bounded by a simple closed path C in the
xy-plane. Use Green’s Theorem to prove that the coordinates
of the centroid sx, yd of D are
x − 1
2A y C
x 2 dy y − 2 1
2A y C
y 2 dx
where A is the area of D.
23. Use Exercise 22 to find the centroid of a quarter-circular
region of radius a.
24. Use Exercise 22 to find the centroid of the triangle with
vertices s0, 0d, sa, 0d, and sa, bd, where a . 0 and b . 0.
25. A plane lamina with constant density sx, yd − occupies a
region in the xy-plane bounded by a simple closed path C.
Show that its moments of inertia about the axes are
I x − 2 3 y C
y 3 dx
I y − 3 y C
x 3 dy
26. Use Exercise 25 to find the moment of inertia of a circular
disk of radius a with constant density about a diameter.
(Compare with Example 15.4.4.)
27. Use the method of Example 5 to calculate y C
F dr, where
Fsx, yd − 2xy i 1 sy2 2 x 2 d j
sx 2 1 y 2 d 2
and C is any positively oriented simple closed curve that
encloses the origin.
28. Calculate y C
F dr, where Fsx, yd − kx 2 1 y, 3x 2 y 2 l and
C is the positively oriented boundary curve of a region D
that has area 6.
29. If F is the vector field of Example 5, show that
y C
F dr − 0 for every simple closed path that does not
pass through or enclose the origin.
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