Ultimately that page is just telling you that the moment of inertia of a rectangle through a given axis is bh^3/12. When determining what is the base and what is the height, the base is parallel to the axis in which you are calculating the moment of inertia from. Sometimes objects will not be such simple shapes like rectangles, squares or circles, so that's why they were showing you all the calculus steps. If you are calculating the moment of inertia of, say a quarter of a circle, after much integrating you're going to get: I_(x or y) = PI*r^4/16 . And say you want to find the moment of inertia about an axis which runs diagonally through a rectangle (for instance the axis runs from the top right to the bottom left corner of the rectangle). Your cartesian components of the moments of inertia again after integrating will become I=bh^3/3.
So those first two examples were pointless. They show us the "proper" way to calculate the moments of inertia of a rectangle. However, we all know that moments of inertia of rectangles = bh^3/12 so those two examples just proved (through integrating) why it was what it was. Note that in these two examples, the shape was centered at the origin.
example 3 just moved the reference axis to an edge of the rectangle, so ultimately u just multiply by 4.
example 4 is like the first two examples, except that you subtract the smaller one from the bigger one.
So if your beam is going to be a rectangle, then to calculate one component of the moment of inertia about an axis which runs straight through this beam should simply be I=bh^3/12 where b is the length of your beam and h is the width of the cross sectional area of your beam. But note that your beam is 3D so you have to account for one more dimension.
But like mentioned, I highly doubt you'll need this
None.