MA36200 Practice Problems #2, Spring 2021
Note:
1. Only one of the five choices (A, B, C, D, E) is correct.
1. A parametrization of the surface x2 + 2y2 + 2z2 = 1 is A .
A. (x, y, z) = (sinφ cos θ, 1√
2
sinφ sin θ, 1√
2
cosφ), where φ ∈ [0, pi] and θ ∈ [0, 2pi]
B. (x, y, z) = (sinφ cos θ, sinφ sin θ, cosφ), where φ ∈ [0, pi] and θ ∈ [0, 2pi]
C. (x, y, z) = (sinφ sin θ, sinφ sin θ, sinφ), where φ ∈ [0, pi] and θ ∈ [0, 2pi]
D. (x, y, z) = (sinφ cos θ, 12 sinφ sin θ,
1
2 cosφ), where φ ∈ [0, pi] and θ ∈ [0, 2pi]
E. none of the above.
2. Let r be the position vector. Then ∇ · r = B
A. 4
B. 3
C. 2
D. 1
E. 0
1
3. ∫ 3
0
∫ 1
−1
9y5exy
3
dydx = C
A. e3 + e−2 + 5
B. e4 − e−3 − 4
C. e3 − e−3 − 6
D. e4
E. e5 − e−5
4. ∫ 2
0
∫ 1
y/2
3yex
3
dxdy = D
A. e
B. e+ 2
C. e2 − 1
D. 2e− 2
E. none of the above.
2
5. Let W denote the region bounded by x = 0, y = 0, z = 0, z = 1 and the cylinder x2 + y2 = 1,
with x > 0 and y > 0. The density distribution of this region is ρ(x, y, z) = z. Then the mass
of this region is E .
A. pi
B. 2pi
C. pi4
D. pi2
E. pi8
6. D denotes the region in the first quadrant of the xy-plane lying between the arcs x2 + y2 = pi
and x2 + y2 = pi/2. The mass density of this region at (x, y) is given by sin(x2 + y2). The total
mass of this region is A .
A. pi/4
B. pi/2
C. pi
D. 2pi
E. 3pi
3
7. Let F(x, y, z) = sin(pix)j− cos(piy)k, and C denotes the triangle with vertices at (1,0,0), (0,1,0)
and (0,0,1) in that order. Then ∫
C
F · ds = B
A. pi − 2
B. 2/pi + 1
C. pi/3 + 2
D. pi2
E. none of the above.
8. The area of the surface defined by z = xy and x2 + y2 6 2 is C .
A. 2pi
B. 2pi/3
C. 2pi3 (3
√
3− 1)
D. pi2 (3
√
3 + 1)
E. none of the above.
4
9. Let S denote the triangle with vertices (1,0,0), (0,2,0) and (0,1,1). The density of the surface
at the point (x, y, z) is xyz. Then the total mass of this surface is D .
A. 1
B.
√
3
C.
√
5
4
D.
√
6
30
E. none of the above.
5

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