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374 CHAPTER 12 Vector Integral Calculus

SECTION 12.1 PROBLEMS

In each of Problems 1 through 10, evaluate the line 7. F · dR with F = xi + yj − zk and C the circle
C
integral. x + y =4, z =0, going around once counterclock-
2
2
wise.

1. xdx − dy + zdz with C given by x = y = t, z = t 3
2
C 8. yz ds with C the parabola z = y , x =1for 0≤ y ≤2
for 0 ≤ t ≤ 1 C √

9. −xyz dz with C given by x = 1, y = z for
2
C
2. −4xdx + y dy − yz dz with C given by x = 4≤ z ≤9
C
2
−t , y = 0, z =−3t for 0 ≤ t ≤ 1
2
10. xz dy with C given by x = y = t, z =−4t for
C
3. (x + y)ds with C given by x = y = t, z = t for 1 ≤ t ≤ 3
2
C
0 ≤ t ≤ 2 11. Find the work done by F = x i − 2yzj + zk in mov-
2
ing an object along the line segment from (1,1,1) to
4. x zds with C the line segment from (0,1,1) to
2
C (4,4,4).
(1,2,−1)
12. Find the mass and center of mass of a thin,

5. F · dR with F = cos(x)i − yj + xzk and R =
C straight wire extending from the origin to (3,3,3) if
2
ti − t j + k for 0 ≤ t ≤ 3 δ(x, y, z) = x + y + z grams per centimeter.
b
6. 4xy ds with C given by x = y = t, z = 2t for 13. Show that any Riemann integral f (x)dx is a line
C a

1 ≤ t ≤ 2 integral F · dR for appropriate choices of F and R.
C
12.2 Green’s Theorem
Green’s theorem is a relationship between double integrals and line integrals around closed
curves in the plane. It was formulated independently by the self-taught amateur British natural
philosopher George Green and the Ukrainian mathematician Michel Ostrogradsky, and is used
in potential theory and partial differential equations.
A closed curve C in the x, y - plane is positively oriented if a point on the curve moves
counterclockwise as the parameter describing C increases. If the point moves clockwise, then C
is negatively oriented. We denote orientation by placing an arrow on the graph, as in Figure 12.3.
A simple closed curve C in the plane encloses a region, called the interior of C.The
unbounded region that remains if the interior is cut out is the exterior of C (Figure 12.4). If
Exterior of C
y
y
Interior of C
x
x

C

FIGURE 12.3 Orientation on a
FIGURE 12.4 Interior and exterior
curve.
of a simple closed curve.

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October 14, 2010 14:53 THM/NEIL Page-374 27410_12_ch12_p367-424