Discussion
The volume of the solid formed by a complete rotation of the shaded portion of the circle of radius R about the y-axis is \(k\pi R^{3}\). The value of k is:
Answer: B
The value is \(k = \frac{5}{24}\).
The radius to the end of the chord makes 60° with the y-axis, so the chord lies at height \(y = R\cos 60^{\circ} = R/2\). The shaded segment runs from \(y = R/2\) to \(y = R\), and rotating it about the y-axis gives a spherical cap. With discs of radius \(x = \sqrt{R^{2} - y^{2}}\):
\[ \begin{aligned} V &= \pi\int_{R/2}^{R}(R^{2} - y^{2})\,dy \\ &= \pi\left[\frac{2R^{3}}{3} - \frac{11R^{3}}{24}\right] = \frac{5}{24}\pi R^{3} \end{aligned} \]
The cap formula \(\pi h^{2}(3R - h)/3\) with \(h = R/2\) gives the same result. 5/12 is twice the correct value, and 7/12 and 7/24 do not follow from a chord at height R/2.
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