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
______________________________________________________________________
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??]
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TEST4/MAC2313
Page 5 of 6
______________________________________________________________________
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
______________________________________________________________________
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
______________________________________________________________________
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.]
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- Spring '06
- GRANTCHAROV
- Multivariable Calculus, Line integral, ρ, Polar coordinate system, 10 pts
-
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