7. (a) Find
$$\int ( 2 x - 1 ) ^ { \frac { 3 } { 2 } } d x$$
giving your answer in its simplest form.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cbfbb690-bc85-46e5-a97f-35df4b6f1c84-13_727_1177_596_370}
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\caption{Figure 3}
\end{figure}
Figure 3 shows a sketch of part of the curve \(C\) with equation
$$y = ( 2 x - 1 ) ^ { \frac { 3 } { 4 } } , \quad x \geqslant \frac { 1 } { 2 }$$
The curve \(C\) cuts the line \(y = 8\) at the point \(P\) with coordinates \(( k , 8 )\), where \(k\) is a constant.
(b) Find the value of \(k\).
The finite region \(S\), shown shaded in Figure 3, is bounded by the curve \(C\), the \(x\)-axis, the \(y\)-axis and the line \(y = 8\). This region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
(c) Find the exact value of the volume of the solid generated.