Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
The solutions to assignment #3 in mth301 (spring 2012) which includes evaluating double integrals using cosine functions, line integrals with respect to arc-length, and applying green's theorem to find the surface integral of a vector field. Topics such as integration, limits, green's theorem, and vector calculus.
Typology: Exercises
1 / 6
This page cannot be seen from the preview
Don't miss anything!
MTH301 (Spring 2012)
Total marks: 10
Lecture # 22 to 37
Question # 1
Evaluate the following double integral
cos 3
0 0
Solution:
cos 3
0 0
4 cos
0 0
4
0
2 2
0
2
0
2
0
2
0 0 0
0 0 0
0 0
0 0
^
Question # 2
Evaluate the following line integral with respect to arc-length s
2 C^1
x ds y
where C is the line x 1 2 , t y t and 0 t 1
Solution:
Here
2
2 3
2
P x xy
and
Q y x y
P x y
Q x y x
Limits for x is from 0 to 2 and for y is 0 to 2
Thus by Green’s Theorem
2 2 2 2 3 2
0 0
2 2 2 2
(^0 ) 2 2
0
3 2 2 0
C
x xy dx y x y dy x y x dy dx
y x xy dx
x x dx
x x
Question # 4
Solution:
Soluition:
The r-limits of integration. A typical ray from the origin
enters R where r = 3 and leaves where r=2+2cos .
Step 3. The -limits of integration.
1
3 cos
cos 1/ 2
cos (1/ 2)
The rays from the origin that intersect R run from
The integral is
/3 2 2
/3 3
COS
f r r drd
^
Question # 5 Find the arc length of parametric curve
2 3 x (1 t ) y (1 t ) 0 t 1
Solution:
dx / dt 2(1 t ), dy / dt 3(1 t )^2
Arc length =
(^1 2 )
0
dx dy dt dt dt
1 2 2 2 0
(^1 2 )
0
t t dt
t t dt
Question # 5
Evaluate the line integral
2 c
I (^) x ydx xdy
where c comprises the three sides of the triangle joining O(0, 0), A (2, 0)
and B (1, 1).
Solution:
(a) OA : c 1 is the line y = 0
Therefore, dy = 0.
2 I 1 (^) (^) cx (0) dx
(b) AB : for c 2 is the line y = -x +
dy = -dx.
