15.4 Green’s Theorem in the Plane, SGLee+김태현
 |
1-4. Use Green’s Theorem to evaluate the line integral 
, when 
is
1. 
, where 
is a circle of 
, with positive orientation.
SKKU-Calculus-Sec- 15-4 Green’s Theorem in Plane, SGLee+김태현
15.4 Green’s Theorem in the Plane, SGLee+김태현
|
1-4. Use Green’s Theorem to evaluate the line integral , when is
1. , where is a circle of , with positive orientation.
2. 
, where 
is the positively oriented curve which is the boundary of the region bounded
by 
, 
.
.
3. 
, where 
is a circle of 
, with positive orientation.
.
let
4. 
where 
is a boundary of the rectangle 
, 
, with positive orientation.
.
5-8. Verify Green's Theorem by evaluating the line integral 
(a) directly, (b) using Green's Theorem, where 
is:
5. 
with 
is a closed curve of the region bounded by 
and 
, with positive orientation.
.
6. 
with 
is a boundary of the region defined by 
and 
, with positive orientation.
.
7. 
with 
is a boundary of the region enclosed by the circles 
and
, with positive orientation.
.
8. 
, where 
is a circle of 
, with positive orientation.
.
9. 
, with positive orientation.
Let 
. To evaluate the line integral, we use Green's Theorem as follows
.
10. 
where 
is the arc of the parabola 
in the plane 
from 
to 
.
http://matrix.skku.ac.kr/cal-lab/cal-15-2-3.html
Using Green's Theorem
11. Calculate the area bounded by the ellipse 
. Hence deduce the area bounded by the circle
.
(a)
.
,
,
Put 
, area of circle : 
.
12. Find the area under arc of the cycloid 
, 
.
13. Find the area under of one arch of the asteroid 
, 
.
14. Find the area of the loop of the folium of Descartes 
, 
.
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15. Let 
be smooth functions satisfying the following differential equations:
, 
. Evaluate the line integral 
,
where 
is traversed in the positive direction of the curve 
.
Using Green's Theorem, we have
.
16. Let 
be a solution of a system of differential equations
, where 
are smooth functions in variables 
.
Assume further that 
is periodic with a periodicity 
,
namely 
for all 
. Suppose that the solution curve (phase trajectory) is
a simple closed curve in the phase plane (
plane) and 
is the region bounded by 
.
Prove that 
.
Proof : Recalling Green's Theorem,
.
Now parametrizing by 
, the line integral becomes
.
Therefore, we obtain
.
17. Evaluate the integral 
, where 
is the boundary of the triangular region 
with corners 
, 
, 
with positive orientation.
18. Let 
be the straight lines from 
to 
, from 
to 
, from 
to 
, and from 
to 
. Evaluate 
.
Let 
. Then
19. Let 
and 
is an any closed curve containing the origin. Find 
.
Let’s consider the region 
which is the annulus as shown in the figure. Let 
and 
denote the outer and the inner circles of 
.
Then by Green’s Theorem, we have
.
Using 
, we can compute
Thus the vector field 
is not conservative. Note that curl
. Here 
is not a simply connected domain.
