3 Fig. 3 shows the curve with parametric equations \(x = t - 3 t ^ { 3 } , y = 1 + 3 t ^ { 2 }\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{07eaad51-dc00-44d2-8bff-8652d62902ec-4_634_1294_388_386}
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\caption{Fig. 3}
\end{figure}
- Show that the values of \(t\) where the curve cuts the \(y\)-axis are \(t = 0 , \pm \frac { 1 } { \sqrt { 3 } }\). Write down the corresponding values of \(y\).
- Find the radius and centre of curvature when \(t = \frac { 1 } { \sqrt { 3 } }\).
The arc of the curve given by \(0 \leqslant t \leqslant \frac { 1 } { \sqrt { 3 } }\) is denoted by \(C\).
- Find the length of \(C\).
- Show that the area of the curved surface generated when \(C\) is rotated about the \(y\)-axis through \(2 \pi\) radians is \(\frac { \pi } { 3 }\).