7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{05b21c5d-5958-4267-b1e6-3d1ed20d5609-25_670_682_301_694}
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\caption{Figure 4}
\end{figure}
A circular tower of radius 1 metre stands in a large horizontal field of grass．A goat is attached to one end of a rope and the other end of the rope is attached to a fixed point \(O\) at the base of the tower．The goat cannot enter the tower．
Taking the point \(O\) as the origin（ 0,0 ），the centre of the base of the tower is at the point \(T ( 0,1 )\) ，where the unit of length is the metre．
The rope has length \(\pi\) metres and you may ignore the size of the goat．
The curve \(C\) shown in Figure 4 represents the edge of the region that the goat can reach．
（a）Write down the equation of \(C\) for \(y < 0\)
When the goat is at the point \(G ( x , y )\) ，with \(x > 0\) and \(y > 0\) ，as shown in Figure 4 ，the rope lies along \(O A G\) where \(O A\) is an arc of the circle with angle \(O T A = \theta\) radians and \(A G\) is a tangent to the circle at \(A\) ．
（b）With the aid of a suitable diagram show that
$$\begin{aligned}
& x = \sin \theta + ( \pi - \theta ) \cos \theta \\
& y = 1 - \cos \theta + ( \pi - \theta ) \sin \theta
\end{aligned}$$
(c) By considering \(\int y \frac { \mathrm {~d} x } { \mathrm {~d} \theta } \mathrm {~d} \theta\), show that the area, in the first quadrant, between \(C\), the positive \(x\)-axis and the positive \(y\)-axis can be expressed in the form
$$\int _ { 0 } ^ { \pi } u \sin u \mathrm {~d} u + \int _ { 0 } ^ { \pi } u ^ { 2 } \sin ^ { 2 } u \mathrm {~d} u + \int _ { 0 } ^ { \pi } u \sin u \cos u \mathrm {~d} u$$
(d) Show that \(\int _ { 0 } ^ { \pi } u ^ { 2 } \sin ^ { 2 } u \mathrm {~d} u = \frac { \pi ^ { 3 } } { 6 } + \int _ { 0 } ^ { \pi } u \sin u \cos u \mathrm {~d} u\)
(e) Hence find the area of grass that can be reached by the goat.