Problem

Figure 3 shows a sketch of the curve $C$ with parametric equations

$$x=2\cos 2t \qquad y=4\sin t \qquad 0\leq t\leq \frac{\pi}{2}$$

The region $R$, shown shaded in Figure 3, is bounded by the curve, the $x$-axis and the $y$-axis.

(a) (i) Show, making your working clear, that the area of $R$ =

$$\int_0^{\frac{\pi}{2}} 32\sin^4 t\cos t\,dt$$

(ii) Hence find, by algebraic integration, the exact value of the area of $R$.

(b) Show that all points on $C$ satisfy $y=\sqrt{ax+b}$, where $a$ and $b$ are constants to be found.

The curve $C$ has equation $y=f(x)$ where $f$ is the function

$$f(x)=\sqrt{ax+b}\qquad -2\leq x\leq 2$$

and $a$ and $b$ are the constants found in part (b).

(c) State the range of $f$.