Surface area of revolution

A question is this type if and only if it asks to find the surface area generated when a parametric curve is rotated about an axis.

2 questions · Challenging +1.2

1.03g Parametric equations: of curves and conversion to cartesian8.06b Arc length and surface area: of revolution, cartesian or parametric
Sort by: Default | Easiest first | Hardest first
Edexcel F3 2023 January Q6
9 marks Challenging +1.2
  1. A curve has parametric equations
    where \(a\) is a positive constant.
$$\begin{aligned} & x = a ( \theta - \sin \theta ) \\ & y = a ( 1 - \cos \theta ) \end{aligned}$$
  1. Show that $$\left( \frac { \mathrm { d } x } { \mathrm {~d} \theta } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} \theta } \right) ^ { 2 } = k a ^ { 2 } \sin ^ { 2 } \frac { \theta } { 2 }$$ where \(k\) is a constant to be determined. The part of the curve from \(\theta = 0\) to \(\theta = 2 \pi\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
  2. Determine the area of the surface generated, giving your answer in terms of \(\pi\) and \(a\).
    [0pt] [Solutions relying on calculator technology are not acceptable.]
Edexcel F3 2015 June Q6
10 marks Challenging +1.2
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0ddee434-f7e1-4f56-91fc-f487112dbf6b-11_709_1269_292_349} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve \(C\) with parametric equations $$x = 2 \cos \theta - \cos 2 \theta , y = 2 \sin \theta - \sin 2 \theta , \quad 0 \leqslant \theta \leqslant \pi$$
  1. Show that $$\left( \frac { \mathrm { d } x } { \mathrm {~d} \theta } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} \theta } \right) ^ { 2 } = 8 ( 1 - \cos \theta )$$ The curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
  2. Find the area of the surface generated, giving your answer in the form \(k \pi\), where \(k\) is a rational number.