- In this question you must show all stages of your working.
You must not use the integration facility on your calculator.
$$I _ { n } = \int t ^ { n } \sqrt { 4 + 5 t ^ { 2 } } \mathrm {~d} t \quad n \geqslant 0$$
- Show that, for \(n > 1\)
$$I _ { n } = \frac { t ^ { n - 1 } } { 5 ( n + 2 ) } \left( 4 + 5 t ^ { 2 } \right) ^ { \frac { 3 } { 2 } } - \frac { 4 ( n - 1 ) } { 5 ( n + 2 ) } I _ { n - 2 }$$
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1241b133-4161-4c04-9b50-067904cc25c2-20_385_394_829_833}
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\caption{Figure 1}
\end{figure}
The curve shown in Figure 1 is defined by the parametric equations
$$x = \frac { 1 } { \sqrt { 5 } } t ^ { 5 } \quad y = \frac { 1 } { 2 } t ^ { 4 } \quad 0 \leqslant t \leqslant 1$$
This curve is rotated through \(2 \pi\) radians about the \(x\)-axis to form a hollow open shell. - Show that the external surface area of the shell is given by
$$\pi \int _ { 0 } ^ { 1 } t ^ { 7 } \sqrt { 4 + 5 t ^ { 2 } } \mathrm {~d} t$$
Using the results in parts (a) and (b) and making each step of your working clear,
- determine the value of the external surface area of the shell, giving your answer to 3 significant figures.