# Solving Density Integral for Solid Cylinder

• -EquinoX-
In summary, 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 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?

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?

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?

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.

The base should be centered at the origin, lying completely in the yz-plane.

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

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?

The shortest distance to the x-axis.

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?

-EquinoX- said:
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?

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?

Yes I believe so. Now do a volume (triple) integration over the region.

well I am confused on how to represent the density in terms of f(x, r, θ)

Didn't you do so above?

one was in terms of x,y and one was in terms of r... but not f(x,r,theta).. I am mostly confused with the theta here

is it just something like this if the cylinder has a radius of 6 ad length 11:

$$\int_0^{11}\int_0^{2\pi} \int_0^6 kr^2 dr d\theta dz$$

Looks ok to me.