% \includegraphics{figure_2} - Shows a circular tower with center T at (0,1), a goat at point G attached to the base at O, with string along arc OA then tangent AG

A circular tower stands in a large horizontal field of grass. A goat is attached to one end of a string and the other end of the string is attached to the fixed point $O$ at the base of the tower. Taking the point $O$ as the origin $(0, 0)$, the centre of the base of the tower is at the point $T(0, 1)$. The radius of the base of the tower is 1. The string has length $\pi$ and you may ignore the size of the goat. The curve $C$ represents the edge of the region that the goat can reach as shown in Figure 2.

(a) Write down the equation of $C$ for $y < 0$.
[1]

When the goat is at the point $G(x, y)$, with $x > 0$ and $y > 0$, as shown in Figure 2, the string lies along $OAG$ where $OA$ is an arc of the circle with angle $OTA = \theta$ radians and $AG$ is a tangent to the circle at $A$.

(b) With the aid of a suitable diagram show that
$$x = \sin \theta + (\pi - \theta) \cos \theta$$
$$y = 1 - \cos \theta + (\pi - \theta) \sin \theta$$
[5]

(c) By considering $\int y \frac{dx}{d\theta} d\theta$, show that the area between $C$, the positive $x$-axis and the positive $y$-axis can be expressed in the form
$$\int_0^{\pi} u \sin u \, du + \int_0^{\pi} u^2 \sin^2 u \, du + \int_0^{\pi} u \sin u \cos u \, du$$
[5]

(d) Show that $\int_0^{\pi} u^2 \sin^2 u \, du = \frac{\pi^3}{6} + \int_0^{\pi} u \sin u \cos u \, du$
[4]

(e) Hence find the area of grass that can be reached by the goat.
[8]

\hfill \mbox{\textit{Edexcel AEA 2014 Q7 [23]}}