\[x = \theta\cos(\theta)\hspace{1cm}y=\theta\sin(\theta)\hspace{1cm}0\leq\theta\leq4\sqrt{3}\]

The spiral object described above exists in the \(xy\) plane and has a linear mass density \((\sigma =\Large{\frac{1}{\theta}})\).

Determine the object's moment of inertia with respect to an axis perpendicular to the \(xy\) plane and passing through the point \((x,y) = (0,0)\).

**Details and Assumptions:** All angles are in radians

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