Problem

Figure 2

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

Figure 2 shows a sketch of the curve C defined by the parametric equations

$$x = \cosh 2t \qquad y = 4\sinh t \qquad 0 \le t \le \frac{3}{2}$$

(a) Show that

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

where $m$ and $n$ are integers to be determined.

(b) Use algebraic integration to determine the exact length of C.

Give the answer in the form $a + \sinh b$ where $a$ and $b$ are integers to be found.

(c) Use algebraic integration to determine the exact area of the surface formed when C is rotated through 360° about the x-axis.

Give the answer in simplest form in terms of hyperbolic functions.