The xy plane that goes from the point 1 1 to the

the xy-plane that goes from the point (-1,-1) to the point (1,1) along the
curve y = x 5 against the force field defined by
F ( x , y ) y x , yx
3
.
W
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8. (10 pts.) Starting at the point (0,0), a particle goes along the
y-axis until it reaches the point (0,4). It then goes from (0,4) to (-4,0)
along the circle with equation x 2 + y 2 = 16. Finally the particle returns
to the origin by travelling along the x-axis. Use Green’s Theorem to
compute the work done on the particle by the force field defined by
F ( x , y ) = < -5 y
3
, 5 x
3
> for ( x , y ) ε
2
. [Draw a picture. This is easy??]

TEST4/MAC2313 Page 5 of 6
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9. (10 pts.) (a) Show that the vector field
F ( x , y ) < cos( x ) e
y
, sin( x ) e
y
10 y >
is actually a gradient field by producing a function φ ( x , y ) such that
∇φ ( x , y ) = F ( x , y ) for all ( x , y ) in the plane.
(b) Using the Fundamental Theorem of Line Integrals, evaluate the path
integral below, where C is any smooth path from the origin to the point
( π /2,ln(2)). [ WARNING: You must use the theorem to get any credit here.]
⌡
⌠
C
(cos( x ) e
y
) dx (sin( x ) e
y
10 y ) dy

TEST3/MAC2313 Page 6 of 6
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10. (10 pts.) Use the substitution u = (1/2)( x + y ) and v = (1/2)( x - y )
to evaluate the integral below, where R is the bound region enclosed by the
triangular region with vertices at (0,0), (2,0), and (1,1).
⌡
⌠
⌡
⌠
R
cos
1
2
( x y ) dA
⌡
⌠
⌡
⌠
R
cos
1
2
( x y ) dA
x , y
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Silly 10 Point Bonus: Become a polar explorer. Reveal the details of
how one can obtain the exact value of the definite integral
⌡
⌠
∞
0
e
x
2
dx
even though there is no elementary anti-derivative for the function
f( x ) e
x 2
.
Say where your work is, for it won’t fit here! [You may gloss some of the
technical details related to the matter of convergence.]