6. (a) A uniform lamina is in the shape of a quadrant of a circle of radius \(a\). Show, by integration, that the centre of mass of the lamina is at a distance of \(\frac { 4 a } { 3 \pi }\) from each of its straight edges.
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\includegraphics[alt={},max width=\textwidth]{daa795f0-2c5e-4617-a295-fbe74c22be4a-10_809_802_484_571}
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\caption{Figure 3}
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A second uniform lamina \(A B C D E F A\) is shown shaded in Figure 3. The straight sides \(A C\) and \(A E\) are perpendicular and \(A C = A E = 2 a\). In the figure, the midpoint of \(A C\) is \(B\), the midpoint of \(A E\) is \(F\), and \(A B D F\) and \(D G E F\) are squares of side \(a\). \(B C D\) is a quadrant of a circle with centre \(B\). \(D G E\) is a quadrant of a circle with centre \(G\).
(b) Find the distance of the centre of mass of the lamina from the side \(A E\).
The lamina is smoothly hinged to a horizontal axis which passes through \(E\) and is perpendicular to the plane of the lamina. The lamina has weight \(W\) newtons. The lamina is held in equilibrium in a vertical plane, with \(A\) vertically above \(E\), by a horizontal force of magnitude \(X\) newtons applied at \(C\).
(c) Find \(X\) in terms of \(W\).