Ellipse (Impossible)
This geometry actually does not contain an analytical solution. But it’s interesting to see why.
Parameterizing the Curve
We parameterize an ellipse similar to how we did the circle. However, there is no nice way to conver it to cylindrical coordinates.
\[\b{\ell}(\phi) = a \cos \phi \; \u{x} + b \sin \phi \; \u{y}\]The full curve ranges from $\phi \in [0, 2 \pi)$.
\[d \b{\ell} = - a \sin \phi \; d\phi \; \u{x} + b \cos \phi \; d\phi \; \u{y}\]Thus,
\[d\ell = \abs{d \b{\ell} } = \sqrt{a^2 \sin^2 \phi + b^2 \cos^2 \phi} \; d\phi\]Mass
\[\begin{align} M &= \int dm \\[10pt] &= \lambda \int d\ell \\[10pt] &= \lambda \int_{0}^{2\pi} \sqrt{a^2 \sin^2 \phi + b^2 \cos^2 \phi} \; d\phi \end{align}\]This integral is actually impossible to evaluate analytically.
Moment of Inertia About Central Axis
\[\begin{align} I &= \int r_{axis}^2 dm \\[10pt] &= \lambda \int r_{axis}^2 d\ell \\[10pt] &= \lambda \int_{0}^{2\pi} (a^2 \sin^2 \phi + b^2 \cos^2 \phi) \sqrt{a^2 \sin^2 \phi + b^2 \cos^2 \phi} \; d\phi \\[10pt] &= \lambda \int_{0}^{2\pi} (a^2 \sin^2 \phi + b^2 \cos^2 \phi)^{3/2} \; d\phi \end{align}\]Again, this integral has no analytical solution.
Moment of Inertia About Central Diameter
\[\begin{align} I &= \int r_{axis}^2 dm \\[10pt] &= \lambda \int r_{axis}^2 d\ell \\[10pt] &= \lambda \int_{0}^{2\pi} (a^2 \sin^2 \phi + b^2 \cos^2 \phi) \sin^2 \phi \sqrt{a^2 \sin^2 \phi + b^2 \cos^2 \phi} \; d \phi \\[10pt] &= \lambda \int_{0}^{2\pi} (a^2 \sin^2 \phi + b^2 \cos^2 \phi)^{3/2} \sin^2 \phi \; d\phi \end{align}\]Likewise here. No solutions :(