Ex1Ch8

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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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)....
View Full Document

## This note was uploaded on 07/19/2010 for the course MA 1C taught by Professor Ramakrishnan during the Spring '08 term at Caltech.

### Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online