Using integration, prove that the centroid ($\bar{y}$) of a rectangle with width $b$ and height $h$ (resting on the x-axis) is located at $h/2$.

Find the centroid location ($\bar{x}, \bar{y}$) of an L-shaped cross-section. The vertical leg is $10\text{ cm}$ tall and $2\text{ cm}$ wide. The horizontal leg extends $8\text{ cm}$ to the right from the bottom of the vertical leg and is also $2\text{ cm}$ thick. (Assume the bottom-left corner of the L is at the origin $(0,0)$).

A semicircular plate has a radius $R = 5\text{ m}$. Its straight edge lies on the x-axis, centered at the origin. What is the location of its centroid?

When loading cargo containers onto a massive ocean freighter, the ship's manifest dictates that heavy containers must be placed deep in the hull, while lighter containers can be stacked high on the deck. Explain this strict loading procedure in terms of the Center of Gravity and rotational stability.

In modern bridge construction, pre-stressed concrete beams are frequently used. The high-tension steel cables run through the beam, but they are almost never placed at the beam's geometric centroid. Why are the cables placed eccentrically (off-center)?