# Density integral

## 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.

## The Attempt at a Solution

Can someone guide me here? is x here a?

## Answers and Replies

Defennder
Homework Helper
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.

how can I get the equation of any point within the circle?

Defennder
Homework Helper
Suppose the point is (x,y). What is distance from the centre to that point?

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?

Defennder
Homework Helper
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.

dx
Homework Helper
Gold Member
The base should be centered at the origin, lying completely in the yz-plane.

dx
Homework Helper
Gold Member
Like this:

http://img22.imageshack.us/img22/5278/terip.jpg [Broken]

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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

dx
Homework Helper
Gold Member
If the coordinates of a point are (x, y, z), what is the distance of this point from the x-axis?

which part of the x-axis? the origin?

dx
Homework Helper
Gold Member
The shortest distance to the x-axis.

is the answer just k(y^2+z^2)?

dx
Homework Helper
Gold Member
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.

hmm.. yea I fogot the sqrt,, how can I express this in terms of r?

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.

polar coordinates... how can I do that?

polar coordinates... how can I do that?

?

I need to represent the density above in terms of r... not y and z

What is r? What does r mean?

I believe it's the radius

Defennder
Homework Helper
Well, you already know the radius is given by $$r=\sqrt{y^2 + z^2}$$. So going from this to an answer in terms of r just requires to replace that latter expression with r.

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?