area moment of inertia explained
The second moment of area, the area moment 2, I = ∫ y² dA, measures how spread an area is about an axis, weighting each piece by the square of its distance from the axis. Because of that square, area placed far from the axis counts enormously, and that is the whole secret of a strong beam. The farther you place it, the steeper the value climbs.
Drag a piece of area away from the axis and the scale changes. The contribution is A d², so doubling the distance multiplies the inertia by four, and distance matters far more than the amount. The same area gives a different result according to where it sits.
For a rectangle about its own centroidal axis, I = b h³12. The depth h enters as a cube, so a little more depth makes it much stiffer, and dragging the depth shows the value shooting up. Growing it taller beats spreading it wider.
Shift to a parallel axis a distance away and the inertia grows by A d². I = Ic + A d², which means the centroidal axis always gives the smallest value, and dragging the distance makes that growth plain. The farther from the center, the penalty sticks on as a square.
The same rectangle has two orientations. Stood on end, the area moves far from the bending axis and the value grows much larger than when it lies flat, so the area is the same while the stiffness differs a lot, and the toggle makes the difference plain. A book stood up bends less than a book laid down.
An I-beam pushes this idea to the extreme. It sends most of the material out to distant flanges and gets a large value from a small area, and toggling against a solid bar of the same area makes the difference obvious. The design empties the inside and keeps the outside.