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dx dy dz 6 6
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 dz dy dx 32 by part (a), since  is constant. Then 2 6 4 x2
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x 0 y 0 z 0
" x " y
Total moment about yz plane Total mass Total moment about xz plane Total mass Total moment about xy plane Total mass
x dz dy dx
x 0 y 0 z 0
Total mass 2 6 4 x2 x 0 y 0 z 0 y dz dy dx 2 Total mass 6 4 x2 z dz dy dx Total mass
24 3 32 4 96 3 32 256=5 8 32 5
z "
x 0 y 0 z 0
Thus, the centroid has coordinates 3=4; 3; 8=5 . " Note that the value for y could have been predicted because of symmetry.
TRANSFORMATION OF TRIPLE INTEGRALS 9.13. Justify equation (11), Page 211, for changing variables in a triple integral.
By analogy with Problem 9.6, we construct a grid of curvilinear coordinate surfaces which subdivide the region r into subregions, a typical one of which is r (see Fig. 9-17).
Fig. 9-17
MULTIPLE INTEGRALS
[CHAP. 9
The vector r from the origin O to point P is r xi yj zk f u; v; w i g u; v; w j h u; v; w k assuming that the transformation equations are x f u; v; w ; y g u; v; w , and z h u; v; w . Tangent vectors to the coordinate curves corresponding to the intersection of pairs of coordinate surfaces are given by @r=@u; @r=@v; @r=@w. Then the volume of the region r of Fig. 9-17 is given approximately by      @r @r @r      u v w  @ x; y; z  u v w @u @v @w @ u; v; w  The triple integral of F x; y; z over the region is the limit of the sum   X  @ x; y; z   Ff f u; v; w ; g u; v; w ; h u; v; w g @ u; v; w  u v w An investigation reveals that this limit is    @ x; y; z   F f f u; v; w ; g u; v; w ; h u; v; w g @ u; v; w  du dv dw
where r is the region in the uvw space into which the region r is mapped under the transformation. Another method for justifying the above change of variables in triple integrals makes use of Stokes theorem (see Problem 10.84, 10).
9.14. What is the mass of a circular cylindrical body represented by 0 @  @ c; 0 @  @ 2; 0 @ z @ h, and with the density function  z sin2 
M h 2 c
0 0 0
region
z sin2  d d dz 
9.15. Use spherical coordinates to calculate the volume of a sphere of radius a.
V 8 a =2 =2
0 0 0
4 a2 sin  dr d d a3 3
9.16. Express
F x; y; z dx dy dz in (a) cylindrical and (b) spherical coordinates.
(a) The transformation equations in cylindrical coordinates are x  cos ; y  sin ; z z. As in Problem 6.39, 6, @ x; y; z =@ ; ; z . Then by Problem 9.13 the triple integral becomes G ; ; z  d d dz
where r is the region in the ; ; z space corresponding to r and where G ; ; z  F  cos ;  sin ; z . (b) The transformation equations in spherical coordinates are x r sin  cos ; y r sin  sin ; z r cos . By Problem 6.101, 6, @ x; y; z =@ r; ;  r2 sin . Then by Problem 9.13 the triple integral becomes H r; ;  r2 sin  dr d d
where r is the region in the r; ;  space corresponding to r, and where H r; ;   F r sin  cos , r sin  sin ; r cos  .
CHAP. 9]
MULTIPLE INTEGRALS
9.17. Find the volume of the region above the xy plane bounded by the paraboloid z x2 y2 and the cylinder x2 y2 a2 .
The volume is most easily found by using cylindrical coordinates. In these coordinates the equations for the paraboloid and cylinder are respectively z 2 and  a. Then Required volume 4 times volume shown in Fig. 9-18 =2 a 2  dz d d 4 4 4 =2 a =2
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