Can somone explain this review solution (moments/inertia)?

[whitejac]

Homework Statement

Homework Equations

This would be the part I think I'm missing

The Attempt at a Solution

In the fourth line, I_x, where is the justification for multiplying 1/3 by the first term? I believe they are using the center of inertia of a rod, which is mr² and the first term doesn't make much sense. Could someone explain to me the justification of that line?

 

[haruspex]

In the fourth line, Ix, where is the justification for multiplying 1/3 by the first term?

There seems to be some confusion over the axes. They must mean I_y.
The part that matters is just the three sides of length 0.2m. Two of them are rotating about one endpoint, hence the 1/3 and the factor 2 at the front. The middle bar is parallel to the axis, so just mr² for that piece.