1. The problem: Explain the Second Pappus’ Theorem, which relates the surface area of a solid of revolution to the centroid of the generating curve. 2. Statement: The Second Pappus’ Theorem states that the surface area $S$ of a solid of revolution generated by rotating a plane curve $C$ about an external axis (not intersecting the curve) is given by: $$S = s \times 2\pi d$$ where $s$ is the arc length of the curve $C$, and $d$ is the distance from the centroid of the curve to the axis of rotation. 3. Important rules: - The curve must be planar and continuous. - The axis of rotation must lie in the same plane but not intersect the curve. - The centroid is the geometric center (center of mass assuming uniform density) of the curve. 4. Explanation: - First, find the centroid $(\bar{x}, \bar{y})$ of the curve $C$ using: $$\bar{x} = \frac{1}{s} \int_C x \, ds, \quad \bar{y} = \frac{1}{s} \int_C y \, ds$$ - Calculate the arc length $s$ of the curve: $$s = \int_C ds$$ - Determine the distance $d$ from the centroid to the axis of rotation. - Multiply the arc length $s$ by the circumference of the circle traced by the centroid, $2\pi d$, to get the surface area. 5. Intuition: As the curve rotates, each point traces a circle. The centroid traces a circle of radius $d$, so the total surface area equals the length of the curve times the circumference of this circle. This theorem simplifies finding surface areas of solids of revolution without performing complex surface integrals.