Let $A$ and $B$ be points on a circle centered at $O$ with radius $R,$ and let $\angle AOB = 2 \alpha \le \pi.$ Minor arc $AB$ is rotated about chord $\overline{AB}.$ Find the surface area of the resulting solid in terms of $R$ and $\alpha.$ This question has been asked before here: Finding the surface area of a solid formed by the rotation of an arc about a chord but it hasn't gotten any answers since it was asked.
Could someone please help me answer it?
Hint...consider finding the equation of a circle radius $R$ whose centre has coordinates $(0,-R\cos\alpha)$.
The section of this circle above the $x$ axis is rotated about the $x$ axis.
If you find $y$ in terms of $x$ you will be able to arrive at $$1+\left(\frac{dy}{dx}\right)^2=\frac{R^2}{R^2-x^2}$$
You can then use the standard integral formula for surface area, with appropriate limits.