Assignment 5 (MATH 215, Q1) 1. Evaluate the triple integral. (a ...

Assignment 5 (MATH 215, Q1)
1. Evaluate the triple integral.
(a)
xy dV , where E is the solid tetrahedron with vertices (0, 0, 0), (1, 0, 0),
E
(0, 2, 0), and (0, 0, 3).
Solution. The plane containing the three points (1, 0, 0), (0, 2, 0), and (0, 0, 3) has an
equation 6x + 3y + 2z = 6. Thus,
E
1 2−2x (6−6x−3y)/2
xydV =
xy dz dy dx
0
0
1 2−2x
=
0
1
=
0
0
0
1
(6 − 6x − 3y)xy dy dx
2
2(−x 4 + 3x 3 − 3x 2 + x) dx = 1
10 .
(b)
E x dV , where E is bounded by the paraboloid x = 4y2 + 4z2 and the plane
x = 4.
Solution. We have Q = {(y, z) : y2 + z2 ≤ 1} and
E
xdV =
0
Q
4
0
4y 2 +4z 2
x dx dA =
Q
8 − 8(y 2 + z 2 ) 2 dA
2π 1
= (8 − 8r 4 1
)r dr dθ = 2π (8r − 8r 5 ) dr = 16π
3 .
2. (a) Find the volume of the region inside the cylinder x 2 + y 2 = 9, lying above the
xy-plane, and below the plane z = y + 3.
Solution. We have Q = {(x, y) : x 2 + y 2 ≤ 9} and
V =
E
dV =
Q
y+3
0
dz dA =
0
Q
(y + 3) dA
2π 3
2π
= (r sin θ + 3) r dr dθ = (9 sin θ + 27/2) dθ = 27π.
0
0
(b) Find the volume of the region bounded by the paraboloids z = x 2 + y 2 and
z = 36 − 3x 2 − 3y 2 .
Solution. We have Q = {(x, y) : x 2 + y 2 ≤ 9} and
V =
E
dV =
Q
36−3x 2 −3y 2
x 2 +y 2
0
dz dA =
Q
(36 − 4x 2 − 4y 2 ) dA
2π 3
= (36 − 4r 2 3
)r dr dθ = 2π (36r − 4r 3 ) dr = 162π.
0
0
1
0

3. Use cylindrical coordinates in the following problems.
(a) Evaluate
E
z = 9 − x2 − y2 and the xy-plane.
x 2 + y 2 dV , where E is the solid bounded by the paraboloid
Solution. In cylindrical coordinates the region E is described by
Thus,
E
x 2 + y 2 dV =
(b) Evaluate the integral
0 ≤ r ≤ 3, 0 ≤ θ ≤ 2π, and 0 ≤ z ≤ 9 − r 2 .
2π 3 2
9−r 2π 3
r · r dz dr dθ = r 2 (9 − r 2 ) dr dθ
0
0
0
0
0
2π 3
= dθ (9r 2 − r 4 ) dθ = 324π
5 .
E x2 dV , where E is the solid that lies within the cylinder
x 2 + y 2 = 1, above the plane z = 0, and below the cone z 2 = 4x 2 + 4y 2 .
Solution. In cylindrical coordinates the region E is described by
Thus,
0 ≤ r ≤ 1, 0 ≤ θ ≤ 2π, and 0 ≤ z ≤ 2r.
E
x 2 2π 1 2r
dV =
(r cos θ) 2 r dz dr dθ
0
0
2π
= cos 2 1
θ dθ
0
0
0
0
0
2r 4 dr = 2π
5 .
4. Use spherical coordinates in the following problems.
(a) Evaluate
dV , where E is the solid that lies between the spheres
E x e(x2 +y 2 +z 2 ) 2
x 2 + y 2 + z 2 = 1 and x 2 + y 2 + z 2 = 4 in the first octant
{(x, y, z) : x ≥ 0, y ≥ 0, z ≥ 0}.
Solution. In spherical coordinates the region E is described by
Thus,
E
x e (x2 +y 2 +z 2 ) 2
1 ≤ ρ ≤ 2, 0 ≤ θ ≤ π/2, 0 ≤ φ ≤ π/2.
π/2 π/2 2
dV =
(ρ sin φ cos θ) e ρ4
ρ 2 sin φ dρ dθ dφ
0
0
π/2
= sin 2 π/2
φ dφ
0
= π
16 (e16 − e).
2
1
0
2
cos θ dθ ρ
1
3 e ρ4
dρ

(b) Evaluate
3
−3
√ 9−x 2
− √ 9−x 2
Solution. The integral is equal to
√ 9−x 2 −y 2
0
z x 2 + y 2 + z 2 dz dy dx.
E z x 2 + y 2 + z 2 dV , where
E = {(ρ, θ, φ) : 0 ≤ ρ ≤ 3, 0 ≤ θ ≤ 2π, 0 ≤ φ ≤ π/2}.
Therefore, the integral is equal to
π/2 2π 3
(ρ cos φ) ρ · ρ 2 sin φ dρ dθ dφ
0
0
0
π/2
2π
= cos φ sin φ dφ
0
0
3
dθ ρ
0
4 dρ = 243π
5 .
5. (a) Find the center of mass of the solid S bounded by the paraboloid z = 4x 2 + 4y 2
and the plane z = 1 if S has constant density K.
Solution. In cylindrical coordinates the region E is described by
Thus, the mass of the solid is
M =
The moment about the xy-plane is
Mxy =
0 ≤ r ≤ 1/2, 0 ≤ θ ≤ 2π, and 4r 2 ≤ z ≤ 1
E
E
2π 1/2 1
K dV =
0
0
4r 2
2π 1/2 1
zK dV =
0
0
4r 2
Kr dz dr dθ = Kπ
8 .
Kz r dz dr dθ = Kπ
12 .
Similarly, the other two moments are Mxz = Myz = 0. We have Mxy/M = 2/3.
Hence, the center of mass is (0, 0, 2/3).
(b) Find the mass of a ball given by x 2 + y 2 + z 2 ≤ a 2 if the density at any point is
proportional to its distance from the z-axis.
Solution. In spherical coordinates the region E is described by
0 ≤ ρ ≤ a, 0 ≤ θ ≤ 2π, 0 ≤ φ ≤ π.
The density is kρ sin φ, where k is a constant. Hence, the mass is
M =
E
2π π a
kρ sin φ dV =
0
0
3
0
(kρ sin φ)ρ 2 sin φ dρ dφ dθ = kπ2 a 4
4
.