Fig. 7 shows the curve BC defined by the parametric equations
$$x = 5 \ln u, \quad y = u + \frac{1}{u}, \quad 1 \leq u \leq 10.$$
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.
\includegraphics{figure_7}
- Find the lengths OA, OB and AC. [5]
- Find \(\frac{dy}{dx}\) in terms of \(u\). Hence find the angle \(\theta\). [6]
- Show that the cartesian equation of the curve is \(y = e^{x/5} + e^{-x/5}\). [2]
An object is formed by rotating the region OACB through \(360°\) about Ox.
- Find the volume of the object. [5]