MTH 277 Graded Assignment 9 (module 18)
Problem 1:
Find

C
ye xy dx  xe xy dy where C is the curve shown below in two ways
a) Verify that F  M i  N j is conservative, find a potential function, and apply the Fundamental
Theorem of Line Integrals.
b) Since F is conservative, you can also replace the path with a simpler path. Do that, and work
directly from the

C
M dx  N dy form. Hint: the straight line direct path from (1, 1) to (3,1)
gives a messy integrand. Easier to break into two paths (and do two integrals); go from (1, 1)
to (3, 1) , and then from (3, 1) to (3,1) .
Problem 2:
Use Green’s Theorem to find

C
F  dr for F  2 xy i  cos(

4
defined by
y2
4 y
x
4
2
2  y  4
Include a sketch of the region.
y ) j and C is the boundary of the region R
Problem 3:
Use Green’s Theorem to find

C
F  dr for F( x, y)  xy 2 i   2 x  y  j and C  C1  C2  C3  C4 is
shown below.
C1 : segment from (0, 4) to (0, 2)
C2 : arc of circle of radius 2 from (0, 2) to ( 2,  2)
C3 : segment from ( 2,  2) to (2 2, 2 2)
C4 : arc of circle of radius 4 from (2 2, 2 2 ) to (0, 4)
One thing to note on this one is that the region is being
traversed clockwise, so you’ll have to switch the sign of the
answer at the end.
For the remaining problems, pick whatever approach you like to evaluate the line integral (apply the
strategy to find the most efficient approach). Approaches may include
1. Parameterize the curves and work straight from
2. Work directly in the

C

C
F  dr
M dx  N dy form
3. Find a potential function and apply the Fundamental Theorem of Integrals
4. Replace the path with a simpler path if appropriate
5. Using Green’s Theorem
Make sure your approach is appropriate for the problem (check for conservative, simply connected
region, etc.).
Problem 4 :

C




 4 y 2  e x dx  2 x  e y dy
2
2
where C is the boundary (traversed counterclockwise) of the rectangle with vertices (0, 0) , (3, 0) ,
(3, 2) and (0, 2) .
Problem 5:
 x
C
2
 y  dx   y 2  x  dy where C  C1  C2 as
shown.
C1 : y  x 3 from (0, 0) to (2,8)
C2 : y  10  x from (2,8) to (4, 6)
Problem 6:

C
F  dr for F( x, y, z )  yz 2 i  xz 2 j  2 xyz k and C is
the curve shown:
Problem 7:

C
F  dr for F( x, y, z )  yz i  xyzj  2 x 2 yz k and C is the space curve given by the parametric
equations
x(t )  2t
y (t )  t  7
z (t )  t 2
0t 4
Problem 8:

C
e x sin( y 2 ) dx  2 ye x cos( y 2 ) dy where C is the path shown below: