A circle of radius \(R\) is centered at the origin. The shaded top portion is bounded above by the circle and below by the horizontal chord through the point where the radius makes \(60^\circ\) with the \(\boldsymbol{y}\)-axis (see figure). The solid formed by a complete rotation of this shaded part about the \(y\)-axis has volume \(k\pi R^{3}\). Find \(k\).
