Problem

The curve $C$ has parametric equations

$$x = \cosh t + t, \qquad y = \cosh t - t, \qquad 0 \le t \le \ln 3.$$

(a) Show that

$$\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 = 2\cosh^2 t.$$

The curve $C$ is rotated through $2\pi$ radians about the $x$-axis. The area of the curved surface generated is given by $S$.

(b) Show that

$$S = 2\pi\sqrt{2}\int_0^{\ln 3} (\cosh^2 t - t\cosh t)\,dt.$$

(c) Hence find the value of $S$, giving your answer in the form

$$\frac{\pi\sqrt{2}}{9}(a + b\ln 3)$$

where $a$ and $b$ are constants to be determined.