Solving Density Integral for Solid Cylinder

[-EquinoX-]

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?

 

[Defennder]

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.

 

[-EquinoX-]

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

 

[Defennder]

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

 

[-EquinoX-]

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]

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.

 

[-EquinoX-]

my picture is not that good, but it's something like this right:

http://img245.imageshack.us/img245/3228/picjef.th.jpg

 

[dx]

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

 

[-EquinoX-]

you mean it's like this:

http://img245.imageshack.us/img245/1531/picz.th.jpg

 

[dx]

Like this:

http://img22.imageshack.us/img22/5278/terip.jpg

 

[-EquinoX-]

ok.. so now I need to find the equation within the circle... I am really confused how this can be achievedf

 

[dx]

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

 

[-EquinoX-]

which part of the x-axis? the origin?

 

[dx]

The shortest distance to the x-axis.

 

[-EquinoX-]

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

 

[dx]

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.

 

[-EquinoX-]

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

 

[Billy Bob]

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.

 

[-EquinoX-]

polar coordinates... how can I do that?

 

[Billy Bob]

polar coordinates... how can I do that?

?

 

[-EquinoX-]

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

 

[Billy Bob]

What is r? What does r mean?

 

[-EquinoX-]

I believe it's the radius

 

[Defennder]

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.

 

[-EquinoX-]

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?

 

[Defennder]

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

 

[-EquinoX-]

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

 

[Defennder]

Didn't you do so above?

 

[-EquinoX-]

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

 

[-EquinoX-]

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

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