Question 5.6: Determine the area of the surface of revolution shown that i...

STRATEGY: According to the first Pappus-Guldinus theorem, the area of the surface of revolution is equal to the product of the length of the arc and the distance traveled by its centroid.
MODELING and ANALYSIS: Referring to Fig. 5.8B and Fig. 1, you have

\bar{x}=2r-\frac{2r}{\pi}=2r\left(1-\frac{1}{\pi} \right) \\ A=2\pi\bar{x}L=2\pi\left[2r\left(1-\frac{1}{\pi} \right) \right] \left(\frac{\pi r}{2} \right)

A = 2πr² (π-1)