Intrinsic equation and curvature

3 The curve \(C\) has parametric equations \(x = 2 t ^ { 3 } - 6 t , y = 6 t ^ { 2 }\).

1. Find the length of the arc of \(C\) for which \(0 \leqslant t \leqslant 1\).
2. Find the area of the surface generated when the arc of \(C\) for which \(0 \leqslant t \leqslant 1\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
3. Show that the equation of the normal to \(C\) at the point with parameter \(t\) is $$y = \frac { 1 } { 2 } \left( \frac { 1 } { t } - t \right) x + 2 t ^ { 2 } + t ^ { 4 } + 3$$
4. Find the cartesian equation of the envelope of the normals to \(C\).
5. The point \(\mathrm { P } ( 64 , a )\) is the centre of curvature corresponding to a point on \(C\). Find \(a\).

3 A curve has parametric equations $$x = a \left( 1 - \cos ^ { 3 } \theta \right) , \quad y = a \sin ^ { 3 } \theta , \quad \text { for } 0 \leqslant \theta \leqslant \frac { \pi } { 3 }$$ where \(a\) is a positive constant.
The arc length from the origin to a general point on the curve is denoted by \(s\), and \(\psi\) is the acute angle defined by \(\tan \psi = \frac { \mathrm { d } y } { \mathrm {~d} x }\).

1. Express \(s\) and \(\psi\) in terms of \(\theta\), and hence show that the intrinsic equation of the curve is $$s = \frac { 3 } { 2 } a \sin ^ { 2 } \psi$$
2. For the point on the curve given by \(\theta = \frac { \pi } { 6 }\), find the radius of curvature and the coordinates of the centre of curvature.
3. Find the area of the curved surface generated when the curve is rotated through \(2 \pi\) radians about the \(y\)-axis.

3 A curve has parametric equations \(x = a ( \theta + \sin \theta ) , y = a ( 1 - \cos \theta )\), for \(0 \leqslant \theta \leqslant \pi\), where \(a\) is a positive constant.

1. Show that the arc length \(s\) from the origin to a general point on the curve is given by \(s = 4 a \sin \frac { 1 } { 2 } \theta\).
2. Find the intrinsic equation of the curve giving \(s\) in terms of \(a\) and \(\psi\), where \(\tan \psi = \frac { \mathrm { d } y } { \mathrm {~d} x }\).
3. Hence, or otherwise, show that the radius of curvature at a point on the curve is \(4 a \cos \frac { 1 } { 2 } \theta\).
4. Find the coordinates of the centre of curvature corresponding to the point on the curve where \(\theta = \frac { 2 } { 3 } \pi\).
5. Find the area of the surface generated when the curve is rotated through \(2 \pi\) radians about the \(x\)-axis.

3 Fig. 3 shows the curve with parametric equations \(x = t - 3 t ^ { 3 } , y = 1 + 3 t ^ { 2 }\). \begin{figure}[h] \end{figure}

1. 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 \(y\).
2. 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 \(C\).
3. Find the length of \(C\).
4. Show that the area of the curved surface generated when \(C\) is rotated about the \(y\)-axis through \(2 \pi\) radians is \(\frac { \pi } { 3 }\).