How Do You Calculate the Moment of Inertia for a Cubic Slab?

[jumbogala]

Homework Statement

A slab has width a, length a, and thickness a/4. What is the moment of inertia about its symmetry axis?

Use the one parallel to the large face of the slab.

Homework Equations

The Attempt at a Solution

The answer is supposedly (ma²)/12 + m(a/4)²/12.

I can't see why so I tried to verify it using integration, but that's not working either. Basically I tried to find the moment of inertia of a thin plate, dimensions a x a. I found that to be ma²/12 like I should.

Then I wanted to basically take a stack of thin plates to make one thick plate of thickness a/4. But I can't figure out what the calculus for that would look like. Help?

 

[SammyS]

Which symmetry axis? It has a couple of them

 

[jumbogala]

If you took a notebook sitting on a table, do the symmetry axis parallel to the table and through the center of the the notebook (horizontal). Since it has dimensions a x a it doesn't matter which direction the axis points (ie. in the x or y direction) because they will be equal.

I just do not want the axis that points up from the table (vertical).

I found something on the internet that shows how to do this, but they did not use a "stack". Is it possible to do it by stacking up a bunch of thin slabs?

 

[SammyS]

The corner to corner diagonal is also a symmetry axis, since it's square.

 

[SammyS]

...

I can't see why so I tried to verify it using integration, but that's not working either. Basically I tried to find the moment of inertia of a thin plate, dimensions a x a. I found that to be ma²/12 like I should.

Then I wanted to basically take a stack of thin plates to make one thick plate of thickness a/4. But I can't figure out what the calculus for that would look like. Help?

It can be done this way using the parallel axis theorem with integration.