7.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{05b21c5d-5958-4267-b1e6-3d1ed20d5609-25_670_682_301_694}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{center}
\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.
\begin{enumerate}[label=(\alph*)]
\item 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 $A$ .
\item With the aid of a suitable diagram show that

$$\begin{aligned}
& x = \sin \theta + ( \pi - \theta ) \cos \theta \\
& y = 1 - \cos \theta + ( \pi - \theta ) \sin \theta
\end{aligned}$$
\item 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

$$\int _ { 0 } ^ { \pi } u \sin u \mathrm {~d} u + \int _ { 0 } ^ { \pi } u ^ { 2 } \sin ^ { 2 } u \mathrm {~d} u + \int _ { 0 } ^ { \pi } u \sin u \cos u \mathrm {~d} u$$
\item Show that $\int _ { 0 } ^ { \pi } u ^ { 2 } \sin ^ { 2 } u \mathrm {~d} u = \frac { \pi ^ { 3 } } { 6 } + \int _ { 0 } ^ { \pi } u \sin u \cos u \mathrm {~d} u$
\item Hence find the area of grass that can be reached by the goat.
\end{enumerate}

\hfill \mbox{\textit{Edexcel AEA 2017 Q7 [25]}}