3 Fig. 7 shows the curve BC defined by the parametric equations
$$x = 5 \ln u , y = u + \frac { 1 } { u } , \quad 1 \leqslant u \leqslant 10$$
The point A lies on the \(x\)-axis and AC is parallel to the \(y\)-axis. The tangent to the curve at C makes an angle \(\theta\) with AC, as shown.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{252453c9-9afa-435c-b64b-5ea37ec69eed-3_505_585_598_766}
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\caption{Fig. 7}
\end{figure}
- Find the lengths \(\mathrm { OA } , \mathrm { OB }\) and AC .
- Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(u\). Hence find the angle \(\theta\).
- Show that the cartesian equation of the curve is \(y = \mathrm { e } ^ { \frac { 1 } { 5 } x } + \mathrm { e } ^ { - \frac { 1 } { 5 } x }\).
An object is formed by rotating the region OACB through \(360 ^ { \circ }\) about \(\mathrm { O } x\).
- Find the volume of the object.