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 $h/2$.

Using integration, determine the centroid coordinates ($\bar{x}$, $\bar{y}$) of a right-angled triangle with base $b$ and height $h$. The right angle is located at the origin $(0,0)$.

Find the centroid location ($\bar{x}, \bar{y}$) of an L-shaped cross-section. The vertical leg is $10.0\text{ cm}$ tall and $2.00\text{ cm}$ wide. The horizontal leg extends $8.00\text{ cm}$ to the right from the bottom of the vertical leg and is also $2.00\text{ cm}$ thick. Assume the bottom-left corner of the L is at the origin $(0,0)$.

A semicircular plate has a radius $R = 5.00\text{ m}$. Its straight edge lies on the x-axis, centered at the origin. What is the location of its centroid?

A solid circular plate of radius $R = 10.0\text{ cm}$ centered at the origin $(0,0)$ has a circular hole of radius $r = 3.00\text{ cm}$ cut out from it. The center of the hole is located at $(5.00\text{ cm}, 0)$. Determine the centroid of the remaining plate.

Find the centroid ($\bar{y}$) of a thin wire bent into a semicircular arc of radius $R = 4.00\text{ cm}$. The wire lies in the upper half of the xy-plane, centered at the origin.

A composite 3D solid consists of a cylinder of radius $R = 2.00\text{ m}$ and height $H = 5.00\text{ m}$, topped by a right circular cone of the same radius $R$ and height $h = 3.00\text{ m}$. Find the z-coordinate of the centroid of the combined volume. The base of the cylinder lies on the xy-plane ($z=0$).

Determine the surface area of a torus (a donut shape) formed by revolving a circle of radius $r = 2.00\text{ cm}$ around an axis located at a distance $R = 5.00\text{ cm}$ from the center of the circle.

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)?

During the launch of a multi-stage rocket, the vehicle consumes thousands of kilograms of propellant per second. How does this massive shift in mass affect the rocket's Center of Gravity, and why is this critical for aerodynamic stability?

In most engineering applications on Earth, the "Center of Mass" and "Center of Gravity" are considered identical. However, for a hypothetical "Space Elevator" stretching from the Earth's equator to $36,000\text{ km}$ into space, these two points would be in vastly different locations. Explain why this divergence occurs.