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Density integral

  • Thread starter -EquinoX-
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  • #1
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Homework Statement



The density of material at any point in a solid cylinder, y^2 + z^2 ≤ a^2, 0 ≤ x ≤ b is proportional to the distance of the point from the x-axis. Find the density. Use k as the proportionality constant.


Homework Equations





The Attempt at a Solution



Can someone guide me here? is x here a?
 

Answers and Replies

  • #2
Defennder
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No x isn't a. That's the parametric representation of a cylinder whose axis coincides with the x-axis. Now from what is given you should be able to tell that the equation describes a circle on the y-z plane. What then is the equation of any point within that circle? That, and using pythagoras theorem should give you the answer.
 
  • #3
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how can I get the equation of any point within the circle?
 
  • #4
Defennder
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Suppose the point is (x,y). What is distance from the centre to that point?
 
  • #5
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the point (x,y) you mean here are you referring that to a point in the circle or.. if I visualize this correctly the cylinder is not standing on it's base.. right?
 
  • #6
Defennder
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Well it depends on what you mean by "base". The base of the cylinder is the y-z plane. So from the conventional orientation of x-y-z plane it's lying on its curved side.
 
  • #8
dx
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The base should be centered at the origin, lying completely in the yz-plane.
 
  • #10
dx
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Like this:

http://img22.imageshack.us/img22/5278/terip.jpg [Broken]
 
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  • #11
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ok.. so now I need to find the equation within the circle... I am really confused how this can be achievedf
 
  • #12
dx
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If the coordinates of a point are (x, y, z), what is the distance of this point from the x-axis?
 
  • #13
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which part of the x-axis? the origin?
 
  • #14
dx
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The shortest distance to the x-axis.
 
  • #15
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is the answer just k(y^2+z^2)?
 
  • #16
dx
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You forgot the square root. The density is k√(y² + z²). You might also want to mention that the density is 0 for √(y² + z²) > a.
 
  • #17
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hmm.. yea I fogot the sqrt,, how can I express this in terms of r?
 
  • #18
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how can I express this in terms of r?
Are you going to use polar/cylindrical coordinates? Use the polar coordinates (r,theta) for the yz plane instead of the usual xy plane.
 
  • #19
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polar coordinates... how can I do that?
 
  • #20
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  • #21
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I need to represent the density above in terms of r... not y and z
 
  • #22
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What is r? What does r mean?
 
  • #23
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I believe it's the radius
 
  • #24
Defennder
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Well, you already know the radius is given by [tex]r=\sqrt{y^2 + z^2}[/tex]. So going from this to an answer in terms of r just requires to replace that latter expression with r.
 
  • #25
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I am asked to set up the integral to find the mass of such a cylinder with radius 6 and length 11.

f(x, r, θ) here is just the density? in terms of x, r and theta?
 

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