Magnetic Moment of a spinning shell

[Ataman]

The problem reads:

Find the magnetic moment of a spinning shell of charge Q, radius R, and angular velocity \vec{\omega}.

My solution:

I split the sphere into infinitely small loops of current (from top to bottom) and add them up. I set my origin on the center of the sphere and integrate with respect to y from 0 to R times two to get the entire sphere.

The start of the problem:

\vec{\mu} \,= A\vec{I}
d\vec{\mu} \,= A(d\vec{I})

Solving for d\vec{I}:

d\vec{I} = \frac{dQ}{t}
d\vec{I} = \frac{\sigma dA}{t}
d\vec{I} = \frac{\sigma dA}{(\frac{2\pi}{\omega})}
d\vec{I} = \frac{\sigma \omega dA}{2\pi}

Plugging d\vec{I} back:

d\vec{\mu} \,= \frac{\sigma \omega A dA}{2\pi}

Changing A and dA, so I could integrate in terms of dy:

A \,= \pi r^2
A \,= \pi (R^2-y^2)
dA \,= -2 \pi y dy

Then setting up the integral:
d\vec{\mu} \,= - \frac{2 \sigma \omega \pi^2}{2\pi} y (R^2-y^2) dy
d\vec{\mu} \,= - \sigma \omega \pi y (R^2-y^2) dy
\vec{\mu} \,= - 2 \sigma \omega \pi \int_{0}^{R} y (R^2-y^2) dy

Without sigma:
\vec{\mu} \,= - \frac {Q \omega}{2R^2} \int_{0}^{R} y (R^2-y^2) dy

This is incorrect, and I have a suspicion it has to do with the "dA" part.

If you don't mind his handwriting, here (http://nebula.deanza.fhda.edu/physics/Newton/4B/MagneticMomentSphere.jpg) is the correct way to do the problem. I am rather uncomfortable integrating over angles, so I opted for a different way.

I would like to know why my way does not work.

-Ataman