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CSS Applied Mathematics QUESTION #1331
Question 1
For the center of mass of a surface of revolution generated by rotating a curve about the \(x\)-axis, the \(x\)-coordinate of the center of mass is given by:
  • \(\bar{x}=\dfrac{\int x\,dS}{\int dS}\) where \(dS\) is the surface area element✔️
  • \(\bar{x}=\dfrac{\int x\,dV}{\int dV}\)
  • \(\bar{x}=\dfrac{\int x\,dA}{\int dA}\) (area)
  • \(\bar{x}=\dfrac{\int x\,ds}{\int ds}\) (arc length only)
Correct Answer Explanation
For a uniform surface of revolution, the center of mass lies on the axis of revolution. The \(x\)-coordinate is \(\bar{x}=\dfrac{\int x\,dS}{\int dS}\) where \(dS=2\pi y\,ds\) is the element of surface area (Pappus's theorem setup). Integration is along the generating curve.