## Transcribed Text

1. (10 pts) Compute
(Hint: Sketch the region on the xy-plane and convert the whole thing into one double integral.)
2. (10 pts) Let D be the region in the first quadrant bounded by x = 0,
y = = y=1-1 as in the figure to the right. Use the
y=1-1
change of variables
(*)
y=
to compute the integral
D
x
(Hint: The inverse map of (*) is given by U = THE and
3. (10 pts) Compute the line integral ScF.dr, where
y
C
x
and C is the portion of the Archimedes' spiral, oriented counterclockwise,
parametrized by
=
(Hint: Split F into the sum of a conservative field and the vortex field.)
5. (10 pts) Use the Green's Theorem to compute the area of the region D in
y
the first quadrant bounded by x = 0, y = 0, and the curve x2/3 +y2/3=
as in the figure to the right.
(Note: You might find S cos2 sin2 tsin4t+Chelpful.) =
D
6. (10 pts) Let C be the curve of intersection of the upper
hemisphere x2 + y2 + z2 = 9, 2 > 0 and the plane Z = y,
oriented clockwise when viewed from above. Compute the line
integral ScF.dr, where
2
= -
1
(Hint: Replace the path of integration to a simpler one using
y
an appropriate theorem.)
-2
2
8. (10 pts) Let S be the boundary of that part of the solid elliptical cylinder y2 + 4z2 16 with
1 0. Use the Divergence Theorem to compute the flux IIs F dS, where
F(x,y,z) = (3xy,3yz,32x). =
9. (10 pts) Let S be the snowman-like closed surface as in the figure to
the right, which is oriented with outward-pointing unit normal. Suppose
you know that the unit ball B = {(x,y,z) : x2 + y2 + z2 < 1} is entirely
contained in the interior of S. Use the Divergence Theorem to compute
the flux given by

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