Math 250B Practice Questions for Test 3

Math 250B
Practice Questions for Test 3
1. Locate the center of mass of the plane region bounded by y = x 2 , x = 0, y = 1 with density
δ(x, y) = xy.
Answer: (x, y) = ( 4
7 , 3 )
4
2. Locate the center of mass of the solid region bounded by x = y 2 , z = 0, x + z = 1 with a
constant density δ(x, y, z) = 1.
Answer: (x, y, z) = ( 3
7 , 0, 2 )
7
3. Set-up an iterated triple integral for ∫∫∫ Q
f(x, y, z) dV where Q is the first-octant region under
the plane of equation 3x + 2y + z = 6.
Answer:
∫ 2 ∫ (6−3x)/2 ∫ 6−3x−2y
0
0
0
f(x, y, z) dz dy dx
4. Set-up a triple integral in cylinder coordinates to compute ∫∫∫ Q (x2 + y 2 ) dV where Q is the
solid region in the first octant that is inside the hemisphere z = √ 9 − x 2 − y 2 and outside the
cylinder x 2 + y 2 = 2y
Answer:
∫ π/2 ∫ 3
0
2 sin θ
∫ √ 9−r 2
0
r 3 dz dr dθ
5. Set-up an iterated triple integral in spherical coordinates to evaluate ∫∫∫ Q (x2 + y 2 ) dV where
Q is the region that lies between the spheres x 2 + y 2 + z 2 = 1 and x 2 + y 2 + z 2 = 9, and inside
the cone z = √ 4x 2 + 4y 2 .
Answer:
∫ 2π ∫ arctan(1/2) ∫ 3
0
0
6. Use spherical coordinates to evaluate
1
∫ 1 ∫ √ 1−x 2
−1
ρ 4 sin 3 φ dρ dφ dθ
∫ √ 2−x 2 −y 2
− √ √
1−x 2 x 2 +y 2
√
x 2 + y 2 + z 2 dz dy dx.
Answer: π(2 − √ 2)
7. Use spherical coordinates to find the mass of the solid region above the cone z = √ x 2 + y 2 and
below z = 3 with density δ = √ x 2 + y 2 + z 2 .
Answer:
∫ 2π ∫ π/4 ∫ 3 sec φ
0
0
0
ρ 3 sin φ dρ dφ dθ = 27π
2 (2√ 2 − 1)
8. Evaluate ∫∫∫ B (x2 + y 2 ) dV where B = {(x, y, z) | x 2 + y 2 + z 2 ≤ 1} is the unit ball in R 3 .
Answer: 8π/15

Math 250B Practice Questions for Test 3 Page 2 of 2
9. Assume that p(x, y, z) is a function that measures the air pollutant particles per cubic meters
of air at any point inside BC Place Stadium in Vancouver. What does
∫∫∫
p(x, y, z) dV
represent physically
(BC Place)
Answer: It represents the total amount of air pollution inside BC Place.
10. Let R be the region bounded by the curves xy = 1, xy = 4, y = 1, and y = 2. Evaluate ∫∫ R exy dA
by using an appropriate change of variables.
Answer: (e 4 − e) ln 2
11.
∫∫
Let R be the parallelogram region with vertices at (0, 0), (3, −3), (5, −2), and (2, 1). Evaluate
(x + y) dA by using an appropriate change of variables.
R
Answer: By using u = x + y and v = x − 2y, we get ∫∫ R
(x + y) dA = 27/2.
Note that if we used (x, y) coordinates, we would need to evaluate three double integrals.
∫ 2 ∫ x/2
0
−x
(x + y) dy dx +
∫ 3 ∫ 3−x
2
−x
(x + y) dy dx +
∫ 5 ∫ 3−x
3
(x−9)/2
(x + y) dy dx = 27
2
12. Let R be the region in the first quadrant bounded by the curves xy = 1, xy = 2, y = x, and
y = 3x. Evaluate ∫∫ R y2 dA by using the change of variables u = xy and v = y/x.
Answer: 3/2