Problem

The curve $C$ has parametric equations

$$x=\cosh t,\qquad y=\sinh t,\qquad\text{for }0<t\le\frac{3}{5}.$$

The length of $C$ is denoted by $s$.

(a) Show that

$$s=\int_0^{\frac{3}{5}}\sqrt{\cosh 2t}\,\mathrm{d}t.$$

(b) By finding the Maclaurin’s series for $\sqrt{\cosh 2t}$ up to and including the term in $t^2$, deduce an approximation to $s$.