Split Disk Moment

Consider a solid, uniform circular disk of mass $1$ and radius $1$ , with its center at $(x,y) = (0,1 )$ . The disk resides within the $xy$ plane.

Each point on the disk can be represented in polar coordinates as $(r,\theta)$ , where $r$ is the distance from the origin (not from the center of the disk) and $\theta$ is the angle with respect to the positive $x$ axis.

Let us move each infinitesimal bit of mass on the disk from $(r,\theta)$ to $(r,\theta + \delta)$ , where $\delta = -\frac{\pi}{4}$ for $(0 \leq \theta \leq \frac{\pi}{2})$ and $\delta = +\frac{\pi}{4}$ for $(\frac{\pi}{2} < \theta \leq \pi)$ . The resulting mass arrangement is as shown in the diagram.

What is the moment of inertia of this split disk with respect to the $y$ axis?

Note: The mass of each infinitesimal portion is preserved as it is moved.