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Ch16Rev Ex 34-37

# Ch16Rev Ex 34-37 - 1044 C H A P T E R 16 M U LTI P L E I N...

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1044 CHAPTER 16 MULTIPLE INTEGRATION (ET CHAPTER 15) x D y 2 Therefore, W is described by W : x 2 + y 2 z 8 ( x 2 + y 2 ), ( x , y ) D Thus, M = ZZ D Z 8 ( x 2 + y 2 ) x 2 + y 2 ( x 2 + y 2 ) 1 / 2 dzdx dy We convert the integral to cylindrical coordinates. The inequalities for W are 0 r 2 , 0 θ 2 π , r 2 z 8 r 2 . Also, ( x 2 + y 2 ) 1 / 2 = r , hence we obtain the following integral: M = Z 2 0 Z 2 0 Z 8 r 2 r 2 r · rdzd dr = Z 2 0 Z 2 0 Z 8 r 2 r 2 r 2 dzd = Z 2 0 Z 2 0 r 2 z ¯ ¯ ¯ ¯ 8 r 2 z = r 2 d = Z 2 0 Z 2 0 r 2 ( 8 r 2 r 2 ) d = Z 2 0 Z 2 0 ( 8 r 2 2 r 4 ) d = ± Z 2 0 1 d Z 2 0 ( 8 r 2 2 r 4 ) ! = 2 ± 8 r 3 3 2 5 r 5 ¯ ¯ ¯ ¯ 2 0 ! = 256 15 53 . 62 34. Describe a region whose volume is equal to: (a) Z 2 0 Z / 2 0 Z 9 4 ρ 2 sin φ d d d (b) Z 1 2 Z / 4 / 3 Z 2 0 rdrd dz (c) Z 2 0 Z 3 0 Z 0 9 r 2 rdzdrd SOLUTION (a) The limits of integration correspond to the inequalities in spherical coordinates, describing the region 4 9 , 0 2 , 0 2 This is, the region between the upper hemispheres of radii 4 and 9. y x z 9 9 9 4 4 4 49 y x (b) The limits of integration correspond to the inequalities in cylindrical coordinates, describing the region W : 0 r 2 , 4 3 , 2 z 1 . The projection of W onto the xy -plane is the sector D shown in the Fgure.

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Chapter Review Exercises 1045 y x 2 1 W is the region above and below D , which is between the planes z =− 2and z = 1.
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Ch16Rev Ex 34-37 - 1044 C H A P T E R 16 M U LTI P L E I N...

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