show that
\end{itemize}
$$\frac { \mathrm { d } A } { \mathrm {~d} \theta } = K ( 1 - \cos \theta )$$
where \(K\) is a constant to be found.
Find, in \(\mathrm { cm } ^ { 2 } \mathrm {~s} ^ { - 1 }\), the rate of increase of the area of the segment when \(\theta = \frac { \pi } { 3 }\)
5.
\begin{figure}[h]
\end{figure}
Figure 2 shows a sketch of the curve defined by the parametric equations
$$x = t ^ { 2 } + 2 t \quad y = \frac { 2 } { t ( 3 - t ) } \quad a \leqslant t \leqslant b$$
where \(a\) and \(b\) are constants.
The ends of the curve lie on the line with equation \(y = 1\)
Find the value of \(a\) and the value of \(b\)
The region \(R\), shown shaded in Figure 2, is bounded by the curve and the line with equation \(y = 1\)
Show that the area of region \(R\) is given by
$$M - k \int _ { a } ^ { b } \frac { t + 1 } { t ( 3 - t ) } \mathrm { d } t$$
where \(M\) and \(k\) are constants to be found.
Write \(\frac { t + 1 } { t ( 3 - t ) }\) in partial fractions.
Use algebraic integration to find the exact area of \(R\), giving your answer in simplest form.