Discussion
The surface generated by revolving \( y = f(x) \) about the \( x \)-axis has area
Answer: C
A band of the surface has circumference \( 2\pi y \) and width equal to the arc element \( ds \), not \( dx \), so
\[ S \;=\; 2\pi\int y\,ds \;=\; 2\pi\int y\sqrt{1 + y'^{2}}\,dx. \]
Using \( dx \) in place of \( ds \) is the standard error: it measures the band along the axis rather than along the slanting surface, and understates the area
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