- Numerical (calculator) integration is not acceptable in this question.
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\caption{Figure 2}
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The shaded region \(O A B\) in Figure 2 is bounded by the \(x\)-axis, the line with equation \(x = 4\) and the curve with equation \(y = \frac { 1 } { 4 } ( x - 2 ) ^ { 3 } + 2\). The point \(A\) has coordinates (4, 4) and the point \(B\) has coordinates \(( 4,0 )\).
A uniform lamina \(L\) has the shape of \(O A B\). The unit of length on both axes is one centimetre. The centre of mass of \(L\) is at the point with coordinates \(( \bar { x } , \bar { y } )\).
Given that the area of \(L\) is \(8 \mathrm {~cm} ^ { 2 }\),
- show that \(\bar { y } = \frac { 8 } { 7 }\)
The lamina is freely suspended from \(A\) and hangs in equilibrium with \(A B\) at an angle \(\theta ^ { \circ }\) to the downward vertical.
- Find the value of \(\theta\).