1. The problem: Explain the First Pappus’ Theorem, which relates the surface area of a solid of revolution to the path traveled by the centroid of the generating curve. 2. Statement of the theorem: The First Pappus’ Theorem states that the surface area $S$ of a solid of revolution generated by rotating a plane curve $C$ about an external axis (in the same plane) is given by: $$S = s \times 2\pi d$$ where: - $s$ is the length of the curve $C$. - $d$ is the distance traveled by the centroid of the curve $C$ during the rotation (i.e., the radius of the centroid's circular path). 3. Important rules and concepts: - The curve $C$ must lie in a plane and be rotated about an axis in the same plane but not intersecting the curve. - The centroid (center of mass) of the curve is the average position of all points on the curve. - The distance $d$ is the radius of the circle traced by the centroid during the rotation. 4. Explanation: - When the curve $C$ is rotated about the axis, every point on $C$ traces a circle. - The surface area generated is the sum of the circumferences of these circles multiplied by infinitesimal lengths along $C$. - The theorem simplifies this by using the total length $s$ of the curve and the path length of the centroid. 5. Formula derivation (conceptual): - The surface area element $dS$ for a small segment $ds$ of the curve is $dS = 2\pi r \, ds$, where $r$ is the distance from the axis to the segment. - Integrating over the curve: $S = \int 2\pi r \, ds$. - By definition of centroid, $d = \frac{\int r \, ds}{s}$. - Thus, $S = 2\pi d s$. 6. Summary: The First Pappus’ Theorem provides a powerful way to find the surface area of solids of revolution by knowing just the length of the generating curve and the path of its centroid. Final answer: $$S = 2\pi d s$$