Stewart_Calc_7ET ch16sec04
MULTIPLE CHOICE
1. Use Green’s Theorem and/or a computer algebra system to evaluate
where C is the circle with counterclockwise orientation.
a.
b.
c.
d.
e.
None of these
2. Let D be a region bounded by a simple closed path C in the xy. Then the coordinates of the
centroid where A is the area of D.
Find the centroid of the triangle with vertices (0, 0), ( , 0) and (0, ).
a.
b.
c.
d.
e.
3. Use Green’s Theorem to evaluate the line integral along the positively oriented closed curve
C.
, where C is the triangle with vertices , , and .
a.
b.
c.
d.
4. Use Green’s Theorem to evaluate the line integral along the positively oriented closed curve
C.
,
where C is the boundary of the region bounded by the parabolas and .
a.
+ e
b.
+ e
c.
d.
5. Use Green’s Theorem to evaluate the line integral along the positively oriented closed curve
C.
, where C is the cardioid .
a.
270
b.
c.
d.
6
60
30
66
−12
5
−6
5
−6
5
−12
5
135
6. Use Green’s Theorem to find the work done by the force
in moving a particle in the positive direction once around the triangle with vertices ,
, and .
a.
7
b.
c.
d.
42
7. A particle starts at the point , moves along the x-axis to (3, 0) and then along the
semicircle to the starting point. Use Green’s Theorem to find the work done on
this particle by the force field
a.
b.
c.
0
d.
e.
8. A plane lamina with constant density occupies a region in the xy-plane
bounded by a simple closed path C. Its moments of inertia about the axes are
Find the moments of inertia about the axes, if C is a rectangle with vertices (0, 0), (4, 0),
(4, 5) and .
a.
b.
c.
7
6
7
3
d.
e.
NUMERIC RESPONSE
1. Use Green’s Theorem to find the work done by the force
in moving a particle from the origin along the x-axis to (1, 0) then along the line segment to
(0, 1) and then back to the origin along the y-axis.
2. Use Green’s Theorem to evaluate the line integral along the given positively oriented curve.
C is the ellipse
3. Evaluate the line integral.
SHORT ANSWER
1. Let R be a plane region of area A bounded by a piecewise-smooth simple closed curve C.
Using Green’s Theorem, it can be shown that the centroid of R is , where
Use these results to find the centroid of the given region.
The triangle with vertices , , and .