Quiz 4 (Section 003): Solutions
Problem 1. Evaluate RR
GRy dV where Gis the solid in the first octant enclosed
by y= 1, y=x,z=x+ 1, and the coordinate planes.
Solution. The solid Gis determined by the inequalities 0 ≤x≤1, x ≤y≤
1,0≤z≤x+ 1. Therefore,
Z Z
GZy dV =
1
Z
0
1
Z
x
x+1
Z
0
y dz dy dx =
1
Z
0
1
Z
x
(x+ 1)y dy dx
=
1
Z
0
(x+ 1)y2
2¯¯
1
y=xdx =
1
Z
0
(x+ 1)1−x2
2dx
=
1
Z
0
x−x3+ 1 −x2
2dx =µx2
4−x4
8+x
2−x3
6¶¯¯
1
0
=1
4−1
8+1
2−1
6=11
24.
Problem 2. Use spherical coordinates to find the mass of the solid bounded
below by the cone z=px2+y2and above by the sphere x2+y2+z2= 9 if its
density is given by δ(x,y , z) = x2+y2+z2.
Solution. The density function in spherical coordinates is given by δ(ρ, θ, φ) =
ρ2. The solid Gin spherical coordinates is determined by the inequalities 0 ≤
θ≤2π, 0≤φ≤π
4,0≤ρ≤3. Thus, the mass of G, calculated in spherical
coordinates, is
M=Z Z
GZδ(ρ, θ, φ)dV =
2π
Z
0
π
4
Z
0
3
Z
0
ρ2ρ2sin φ dρ dφ dθ
=
2π
Z
0
π
4
Z
0µρ5
5sin φ¶¯¯
3
ρ=0 dφ dθ =
2π
Z
0
π
4
Z
0
243
5sin φ dφ dθ
=
2π
Z
0µ−243
5cos φ¶¯¯
π
4
φ=0 dθ =−2π243
5Ã√2
2−1!
=243π
5(2 −√2).
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