554 CHAPTER 27. THE INTEGRAL IN OTHER COORDINATES

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.

9. 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 x axis given the density equals 1.

10. Let B be a solid ball of constant density and radius R. Find the moment of inertiaabout a line through a diameter of the ball. You should get 2

5 R2M where M is themass..

11. Let B be a solid ball of density σ = ρ where ρ is the distance to the center of the ballwhich has radius R. Find the moment of inertia about a line through a diameter ofthe ball. Write your answer in terms of the total mass and the radius as was done inthe constant density case.

12. Let C be a solid cylinder of constant density and radius R. Find the moment of inertiaabout the axis of the cylinder

You should get 12 R2M where M is the mass.

13. Let C be a solid cylinder of constant density and radius R and mass M and let B be asolid ball of radius R and mass M. The cylinder and the ball are placed on the top ofan inclined plane and allowed to roll to the bottom. Which one will arrive first andwhy?

14. A ball of radius 4 has a cone taken out of the top which has an angle of π/2 and thena cone taken out of the bottom which has an angle of π/3. If the density is λ = ρ ,find the z component of the center of mass.

15. A ball of radius 4 has a cone taken out of the top which has an angle of π/2 and thena cone taken out of the bottom which has an angle of π/3. If the density is λ = ρ ,find the moment of inertia about the z axis.