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QUIZ 4, Version A, MATH 251, Section 505
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Write up your result, detail your calculations if necessary and BOX your final answer
1. [50pts] Evaluate the work of the force F (x, y) = h2xy, y 2 i along the parabola y = x2 + 1 from (−1, 2)
to (1, 2).
2. [50pts] Compute the line integral
R
C
x2 yds where C is the first quadrant of the circle x2 + y 2 = 2.
R
1. The work of F along the part of the parabola denoted, C is C F.dr.
Let’s parametrize C, r(t) =< t, t2 + 1 > for increasing t from −1 to 1.
Z
Z
1
1
F (r(t)).r (t)dt =
F.dr =
C
Z
0
< 2t(t2 + 1), (t2 + 1)2 > . < 1, 2t > dt,
−1
−1
Z 1
(t2 + 1)3 (t2 + 1)2
=
2t(t + 1) + 2t(t + 1) dt =
+
3
2
−1
2
2
2
1
,
−1
= 0.
2. It’s
with respect the arc length. The part of
√
√ an integral
√ the circle C is parametrized by r(t) =<
π
0
2 cos t, 2 sin t >, 0 ≤ t ≤ 2 . We have ds = |r (t)|dt = 2dt,
Z
Z
2
y xds =
C
π
2
0
=
4
.
3
cos3 t
2.2 cos t sin tdt = 4 −
3
2
π2
,
0