Find Moment of Inertia Around CoM: Summation Formula & Point Mass

[genericpumpkin]

How do I find the moment of inertia around the CoM of an object when the axis of rotation is not through the CoM?

When Are summation formula used in equations and what exactly constitutes a point mass? regarding moments of inertia?

 

[SteamKing]

Use the parallel axis theorem to transfer the MOI from the c.o.m. to the axis of rotation.

 

[BiGyElLoWhAt]

I took the concept of a parallel axis shift and didn't require it to be parallel.

DoubleIntegral from -r_y to(L_y-r_y) and -r_x to (L_x - r_x) of ((x-r_x)^2+(y-r_y)^2)dm
For dm I used Mdxdy/(L_x)(L_y)

Clarifying, x and y are my variables, L_x is the x length of my object, and L_y is the length in the y direction. r_x and r_y are the distances or "shifts" of my axis from the center of mass in the x and y directions respectively.

Here's what I got (after quite a bit of simplification)

M [ (L_x^2)/3 - 2(L_x)(r_x) +3r_x^2 + (L_y^2)/3 - 2(L_y)(r_y) + 3r_y^2]

Anyone care to check my work? Or at least my initial setup? I didn't really get as much out of moments as I could've in my mechanics class, so I took this opportunity to brush up on them.
Btw I'm integrating a uniform density rectangular area moment. I hope you followed that lol