7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{08f62966-2e63-4542-a10a-c6453a3215e7-10_542_1164_251_477}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
Figure 2 shows a sketch of the curve \(C\) with parametric equations
$$x = 27 \sec ^ { 3 } t , y = 3 \tan t , \quad 0 \leqslant t \leqslant \frac { \pi } { 3 }$$
- Find the gradient of the curve \(C\) at the point where \(t = \frac { \pi } { 6 }\)
- Show that the cartesian equation of \(C\) may be written in the form
$$y = \left( x ^ { \frac { 2 } { 3 } } - 9 \right) ^ { \frac { 1 } { 2 } } , \quad a \leqslant x \leqslant b$$
stating the values of \(a\) and \(b\).
(3)
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{08f62966-2e63-4542-a10a-c6453a3215e7-10_581_1173_1628_475}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
The finite region \(R\) which is bounded by the curve \(C\), the \(x\)-axis and the line \(x = 125\) is shown shaded in Figure 3. This region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution. - Use calculus to find the exact value of the volume of the solid of revolution.