Problem

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

Solutions relying entirely on calculator technology are not acceptable.

A curve $C$ has parametric equations

$$x=\sin^2 t\qquad y=2\tan t\qquad 0\le t<\frac{\pi}{2}$$

The point $P$ with parameter $t=\dfrac{\pi}{4}$ lies on $C$.

The line $l$ is the normal to $C$ at $P$, as shown in Figure 3.

(a) Show, using calculus, that an equation for $l$ is

$$8y+2x=17$$

The region $S$, shown shaded in Figure 3, is bounded by $C$, $l$ and the $x$-axis.

(b) Find, using calculus, the exact area of $S$.