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The integration with respect to z (keeping  and  constant) from z 0 to z 2 corresponds to summing the cubical volumes (indicated by dV in a vertical column extending from the xy plane to the paraboloid. The subsequent integration with respect to  (keeping  constant) from  0 to  a corresponds to addition of volumes of all columns in the wedge-shaped region. Finally, integration with respect to  corresponds to adding volumes of all such wedge-shaped regions. The integration can also be performed in other orders to yield the same result. We can also set up the integral by determining the region r 0 in ; ; z space into which r is mapped by the cylindrical coordinate transformation.
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9.18. (a) Find the moment of inertia about the z-axis of the region in Problem 9.17, assuming that the density is the constant . (b) Find the radius of gyration.
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(a) The moment of inertia about the z-axis is =2 a
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2
4
=2 a
 0 z 0
2  dz d d =2  6  6 a  d a   3  0 6  0
 0  0
5 d d 4
MULTIPLE INTEGRALS
[CHAP. 9
The result can be expressed in terms of the mass M of the region, since by Problem 9.17, M volume density  4 a  2 so that Iz a6  a6 2M 2 Ma2 3 3 a4 3
Note that in setting up the integral for Iz we can think of  dz d d as being the mass of the cubical volume element, 2  dz d d, as the moment of inertia of this mass with respect to the z-axis and 2  dz d d as the total moment of inertia about the z-axis. The limits of integration are determined as in Problem 9.17. p (b) The radius of gyration is the value K such that MK 2 2 Ma2 , i.e., K 2 2 a2 or K a 2=3. 3 3 The physical signi cance of K is that if all the mass M were concentrated in a thin cylindrical shell of radius K, then the moment of inertia of this shell about the axis of the cylinder would be Iz .
9.19. (a) Find the volume of the region bounded above by the sphere x2 y2 z2 a2 and below by the cone z2 sin2 x2 y2 cos2 , where is a constant such that 0 @ @ . (b) From the result in (a), nd the volume of a sphere of radius a.
In spherical coordinates the equation of the sphere is r a and that of the cone is  . This can be seen directly or by using the transformation equations x r sin  cos ; y r sin  sin , z r cos . For example, z2 sin2 x2 y2 cos2 becomes, on using these equations, r2 cos2  sin2 r2 sin2  cos2  r2 sin2  sin2  cos2 i.e., r2 cos2  sin2 r2 sin2  cos2 from which tan  tan and so  or   . It is su cient to consider one of these, say,  . a Required volume 4 times volume (shaded) in Fig. 9-19 =2 a 4 r2 sin  dr d d   r sin  d d   0  0 3 r 0 3 =2 4a sin  d d 3  0  0  =2  4a3 cos  d  3  0  0 4 =2 2a3 1 cos 3
 0  0 r 0 3
Fig. 9-19
The integration with respect to r (keeping  and  constant) from r 0 to r a corresponds to summing the volumes of all cubical elements (such as indicated by dV) in a column extending from r 0 to r a. The subsequent integration with respect to  (keeping  constant) from  0 to  =4 corresponds to summing the volumes of all columns in the wedge-shaped region. Finally, integration with respect to  corresponds to adding volumes of all such wedge-shaped regions.
CHAP. 9]
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