9.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{66abdef1-072e-41eb-a933-dd51a96330ff-24_803_1050_251_511}
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\caption{Figure 3}
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In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
Figure 3 shows a sketch of part of the curve \(C\) with equation
$$y = \frac { 2 } { 3 } x ^ { 2 } - 9 \sqrt { x } + 13 \quad x \geqslant 0$$
- Find, using calculus, the range of values of \(x\) for which \(y\) is increasing.
The point \(P\) lies on \(C\) and has coordinates (9, 40).
The line \(l\) is the tangent to \(C\) at the point \(P\).
The finite region \(R\), shown shaded in Figure 3, is bounded by the curve \(C\), the line \(l\), the \(x\)-axis and the \(y\)-axis. - Find, using calculus, the exact area of \(R\).