Ex1Ch8

Ex1Ch8 - [(4 / 3) π ] / 16 = π/ 12 . (b) The required...

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1 Review Example 1, Chapter 8. Let W be the region in the octant x 0 ,y 0 ,z 0 , bounded by the three planes y = 0 ,z = 0 ,x = y , and by the sphere x 2 + y 2 + z 2 = 1 . (a) Find the volume of W . (b) Set up a triple integral giving the integral of a function f ( x,y,z ) over this region using spherical coordinates. (c) Calculate the surface integral ZZ S F · d S where F ( x,y,z ) = (3 x - z 4 ) i - ( x 2 - y ) j + ( xy 2 ) k and S is the boundary of the set W . Solution. First we draw the following figure. y z x = y x W (a) The volume of the region W is 1 / 16th that of a unit sphere, so it is
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Unformatted text preview: [(4 / 3) π ] / 16 = π/ 12 . (b) The required integral is Z π/ 2 φ =0 Z π/ 4 θ =0 Z 1 ρ =0 f ( ρ sin φ cos θ,ρ sin φ sin θ,ρ cos φ ) ρ 2 sin φdρdθ dφ by the change of variables formula. 2 (c) Notice that div F = 4, so that by the divergence theorem, ZZ S F · d S = ZZZ W div F dxdy dz = ZZZ W 4 dxdy dz = 4 × vol( W ) = π/ 3 using part (a)....
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This note was uploaded on 07/19/2010 for the course MA 1C taught by Professor Ramakrishnan during the Spring '08 term at Caltech.

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Ex1Ch8 - [(4 / 3) π ] / 16 = π/ 12 . (b) The required...

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