1. Let R be the region in the xy-plane bounded between the circle x2 + y 2 = 4 and the circle
x2 + y 2 = 9.
(5 points)
(a) Sketch the region R. Label each axis with appropriate units.
y
x
(8 points)
(b) Compute the double integral
! "
x2 + y 2 dA.
R
(11 points) 2. On the axes below, sketch the gradient field for the function f (x, y) = x2 + y 2 + 2.
y
x
(11 points) 3. Find a parametrization for the circle of radius 3 parallel to the xy-plane, centered at the point
(0, 0, 2).
3.
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4. A ball is pushed off a table at time t = 0 seconds. Its position at time t ≥ 0 is given by
!r(t) = t!i + 2t!j + (4 − 16t2 )!k,
where the origin is at the base of the table, standing on flat ground. Distance is measure in
feet. The vector !i points east, !j points north, and !k points up.
(6 points)
(a) At what time does the ball hit the ground?
(a)
(7 points)
(b) How fast is the ball moving when it hits the ground? Include appropriate units.
(b)
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(13 points) 5. Write but do not evaluate a triple integral in cylindrical coordinates that gives the volume
of the!region inside the cylinder x2 + y 2 = 4 which lies above the downward-facing cone
z = − x2 + y 2 and below the plane z = x + y + 5.
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!
(13 points) 6. Compute the line integral C F! · d!r, where F! = y!i − x!j, and C is the right-hand side of the
unit circle, traversed from (0, −1) to (0, 1).
6.
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(13 points) 7. Find the flow line of the vector field F! (x, y) = y!i + 4!j that passes through the point (5, 6) at
time t = 1.
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(13 points) 8. Use spherical coordinates to evaluate the following triple integral:
!
0
1!
√
−
1−x2
√
1−x2
!
√
−
1−x2 −z 2
√
1−x2 −z 2
1
dy dz dx.
(x2 + y 2 + z 2 )1/2
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