Problem

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

Solutions relying entirely on calculator technology are not acceptable.

Figure 2 shows a sketch of part of the curve $C$ with equation $y^2=8x$ and part of the line $l$ with equation $x=18$

The region $R$, shown shaded in Figure 2, is bounded by $C$ and $l$

(a) Show that the perimeter of $R$ is given by

$$\alpha+2\int_0^{\beta}\sqrt{1+\frac{y^2}{16}}\,dy$$

where $\alpha$ and $\beta$ are positive constants to be determined.

(b) Use the substitution $y=4\sinh u$ and algebraic integration to determine the exact perimeter of $R$, giving your answer in simplest form.