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where we have used the results of Problem 9.7.
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[CHAP. 9
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Fig. 9-12 Note that the limits of integration for the region r 0 can be constructed directly from the region r in the xy plane without actually constructing the region r 0 . In such case we use a grid as in Problem 9.6. The coordinates u; v are curvilinear coordinates, in this case called hyperbolic coordinates.
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q 9.9. Evaluate x2 y2 dx dy, where r is the region in the xy plane bounded by x2 y2 4 and x2 y2 9.
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The presence of x2 y2 suggests the use of polar coordinates ;  , where x  cos ; y  sin  (see Problem 6.39, 6). Under this transformation the region r [Fig. 9-13(a) below] is mapped into the region r 0 [Fig. 9-13(b) below].
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Fig. 9-13 Since @ x; y , it follows that @ ;   q q   @ x; y   d d   d d x2 y2 dx dy x2 y2  @ ;  
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CHAP. 9]
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We can also write the integration limits for r 0 immediately on observing the region r, since for xed ,  varies from  2 to  3 within the sector shown dashed in Fig. 9-13(a). An integration with respect to  from  0 to  2 then gives the contribution from all sectors. Geometrically,  d d represents the area dA as shown in Fig. 9-13(a).
9.10. Find the area of the region in the xy plane bounded by the lemniscate 2 a2 cos 2.
Here the curve is given directly in polar coordinates ;  . By assigning various values to  and nding corresponding values of , we obtain the graph shown in Fig. 9-14. The required area (making use of symmetry) is =4 3 ap =4 ap cos 2   cos 2   d d 4 d 4   0  0  0 2  0 =4 =4  a2 cos 2 d a2 sin 2 a2 2 
 0  0
Fig. 9-14
Fig. 9-15
TRIPLE INTEGRALS 9.11. (a) (b) Sketch the three-dimensional region r bounded by x y z a a > 0 ; x 0; y 0; z 0. Give a physical interpretation to x2 y2 z2 dx dy dz
Evaluate the triple integral in (b).
(a) The required region r is shown in Fig. 9-15. (b) Since x2 y2 z2 is the square of the distance from any point x; y; z to 0; 0; 0 , we can consider the triple integral as representing the polar moment of inertia (i.e., moment of inertia with respect to the origin) of the region r (assuming unit density). We can also consider the triple integral as representing the mass of the region if the density varies as x2 y2 z2 .
MULTIPLE INTEGRALS
[CHAP. 9
The triple integral can be expressed as the iterated integral a
a x a x y
y 0 z 0
x2 y2 z2 dz dy dx  z3 a x y x2 z y2 z  dy dx 3 z 0 x 0 y 0 ( a a x
a a x 2 2
) a x y 3 dy dx x a x x y a x y y 3 x 0 y 0  a x2 y2 a x y3 y4 a x y 4 a x  dx x2 a x y  2 3 4 12 x 0 y 0 ) a( 2 2 4 4 x a x a x a x a x 4 x2 a x 2 dx 2 3 4 12 0 ) a( 2 x a x 2 a x 4 a5 dx 2 6 20 0
The integration with respect to z (keeping x and y constant) from z 0 to z a x y corresponds to summing the polar moments of inertia (or masses) corresponding to each cube in a vertical column. The subsequent integration with respect to y from y 0 to y a x (keeping x constant) corresponds to addition of contributions from all vertical columns contained in a slab parallel to the yz plane. Finally, integration with respect to x from x 0 to x a adds up contributions from all slabs parallel to the yz plane. Although the above integration has been accomplished in the order z; y; x, any other order is clearly possible and the nal answer should be the same.
9.12. Find the (a) volume and (b) centroid of the region r bounded by the parabolic cylinder z 4 x2 and the planes x 0, y 0, y 6, z 0 assuming the density to be a constant .
The region r is shown in Fig. 9-16.
Fig. 9-16
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