Moment of inertia of a rotated line

[rock.freak667]

Homework Statement

I have a drum that comprises of basically a cylinder and rotated line in the form y=mx+c

http://img22.imageshack.us/img22/6108/druma.jpg

Homework Equations

$$I=\int r^2 dm$$

The Attempt at a Solution

I found the MOI for the cylinder as 1/2M(a²+b²) where a and b are the inner and outer radii respectively.

I am getting trouble forming the MOI for the line. I am supposed to consider an elemental section, but I am not sure if I should consider a disc or some other shape.

Disc: (Width dx and radius 'y')

dI=1/2(dm)y²

dm=ρ dV = ρ(πy²dx)

dI=1/2ρπ y⁴ dx

$$I_{line}= \frac{1}{2} \rho \pi \int_{x_1} ^{x_2} y^4 dx$$


Since I have two lines, the inertia would be


$$I = \frac{1}{2} \rho \pi \int_{x_1} ^{x_2} (y_2^4 -y_1^4) dx$$

I can appropriately select x₂ and x₁ based on height of the cylinder and the height of the total drum. I can find the ρ for the material. But I'd need to calculate the equations for y₂ and y₁.

So my MOI would be

$$I = \frac{1}{2} M(a^2+b^2)+ \frac{1}{2} \rho \pi \int_{x_1} ^{x_2} (y_2^4 -y_1^4) dx$$

Is this correct or did I make a mistake in my element or in the algebra?