9.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{59c9f675-e7eb-47b9-b233-dfbe1844f792-30_639_929_214_511}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
Figure 2 shows
- the curve \(C\) with equation \(y = x - x ^ { 2 }\)
- the line \(l\) with equation \(y = m x\), where \(m\) is a constant and \(0 < m < 1\)
The line and the curve intersect at the origin \(O\) and at the point \(P\).
- Find, in terms of \(m\), the coordinates of \(P\).
The region \(R _ { 1 }\), shown shaded in Figure 2, is bounded by \(C\) and \(l\).
- Show that the area of \(R _ { 1 }\) is
$$\frac { ( 1 - m ) ^ { 3 } } { 6 }$$
The region \(R _ { 2 }\), also shown shaded in Figure 2, is bounded by \(C\), the \(x\)-axis and \(l\). Given that the area of \(R _ { 1 }\) is equal to the area of \(R _ { 2 }\)
- find the exact value of \(m\).
\includegraphics[max width=\textwidth, alt={}, center]{59c9f675-e7eb-47b9-b233-dfbe1844f792-33_108_76_2613_1875}
\includegraphics[max width=\textwidth, alt={}, center]{59c9f675-e7eb-47b9-b233-dfbe1844f792-33_52_83_2722_1850}