Derive the expression for the moment of inertia of a rectangle of base $b$ and height $h$ about an axis passing through its centroid and parallel to its base ($I_{xc}$).

A wooden joist has a rectangular cross-section with width $b = 50\text{ mm}$ and height $h = 200\text{ mm}$. Calculate the moment of inertia about its base (the bottom edge).

Calculate the moment of inertia $I_x$ about the centroidal horizontal axis for a symmetrical steel I-beam. The top and bottom flanges are $200\text{ mm}$ wide and $20\text{ mm}$ thick. The web is $300\text{ mm}$ tall and $10\text{ mm}$ thick.

Why is the I-beam (or Wide Flange section) the ubiquitous shape for structural steel girders in skyscrapers and bridges, rather than solid rectangular or circular bars of the same weight? Explain using the concept of the Moment of Inertia.

Automotive drive shafts are almost always hollow tubes rather than solid steel rods, even though solid rods are nominally "stronger." Why is a hollow tube preferred for transmitting torque? Explain using the Polar Moment of Inertia ($J$).