20.8. EXERCISES 385
Example 20.7.3 Find the moment of inertia about the z axis and center of mass of the solidwhich occupies the region R defined by 9−
(x2 + y2
)≥ z ≥ 0 if the density is ρ (x,y,z) =√
x2 + y2.
This moment of inertia is∫
R(x2 + y2
)√x2 + y2 dV and the easiest way to find this
integral is to use cylindrical coordinates. Thus the answer is
∫ 2π
0
∫ 3
0
∫ 9−r2
0r3r dzdr dθ =
874835
π.
To find the center of mass, note the x and y coordinates of the center of mass,∫R xρ dV∫R ρ dV
,
∫R yρ dV∫R ρ dV
both equal zero because the above shape is symmetric about the z axis and ρ is also sym-metric in its values. Thus xρ dV will cancel with −xρ dV and a similar conclusion willhold for the y coordinate. It only remains to find the z coordinate of the center of mass, z.In polar coordinates, ρ = r and so,
z =∫
R zρ dV∫R ρ dV
=
∫ 2π
0∫ 3
0∫ 9−r2
0 zr2 dzdr dθ∫ 2π
0∫ 3
0∫ 9−r2
0 r2 dzdr dθ
=187.
Thus the center of mass will be(0,0, 18
7
).
20.8 Exercises1. Let R denote the finite region bounded by z = 4− x2− y2 and the xy plane. Find zc,
the z coordinate of the center of mass if the density σ is a constant.
2. Let R denote the finite region bounded by z = 4− x2− y2 and the xy plane. Find zc,the z coordinate of the center of mass if the density σ is equals σ (x,y,z) = z.
3. Find the mass and center of mass of the region between the surfaces z =−y2 +8 andz = 2x2 + y2 if the density equals σ = 1.
4. Find the mass and center of mass of the region between the surfaces z =−y2 +8 andz = 2x2 + y2 if the density equals σ (x,y,z) = x2.
5. The two cylinders, x2 + y2 = 4 and y2 + z2 = 4 intersect in a region R. Find the massand center of mass if the density σ , is given by σ (x,y,z) = z2.
6. The two cylinders, x2 + y2 = 4 and y2 + z2 = 4 intersect in a region R. Find the massand center of mass if the density σ , is given by σ (x,y,z) = 4+ z.
7. Find the mass and center of mass of the set (x,y,z) such that x2
4 + y2
9 + z2 ≤ 1 if thedensity is σ (x,y,z) = 4+ y+ z.
8. Let R denote the finite region bounded by z = 9− x2− y2 and the xy plane. Find themoment of inertia of this shape about the z axis given the density equals 1.