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The diagram shows the curve \(C\) with parametric equations
$$x = \frac { 1 } { 4 } \sin t , \quad y = t \cos t$$
where \(0 \leqslant t \leqslant k\).
- Find the value of \(k\).
- Find \(\frac { \mathrm { d } y } { \mathrm {~d} t }\) in terms of \(t\).
The maximum point on \(C\) is denoted by \(P\).
- Using your answer to part (ii) and the standard small angle approximations, find an approximation for the \(x\)-coordinate of \(P\).
- (a) Show that the area of the finite region bounded by \(C\) and the \(x\)-axis is given by
$$b \int _ { 0 } ^ { a } t ( 1 + \cos 2 t ) \mathrm { d } t$$
where \(a\) and \(b\) are constants to be determined.
(b) In this question you must show detailed reasoning.
Hence find the exact area of the finite region bounded by \(C\) and the \(x\)-axis.