Problem

Figure 2 shows a sketch of the curve with parametric equations

$$x=\sqrt{9-4t}\qquad y=\frac{t^3}{\sqrt{9+4t}}\qquad 0\leq t\leq \frac{9}{4}$$

The curve touches the $x$-axis when $t=0$ and meets the $y$-axis when $t=\dfrac{9}{4}$

The region $R$, shown shaded in Figure 2, is bounded by the curve, the $x$-axis and the $y$-axis.

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

$$K\int_0^{\frac{9}{4}} \frac{t^3}{\sqrt{81-16t^2}}\,dt$$

where $K$ is a constant to be found.

(b) Using the substitution $u=81-16t^2$, or otherwise, find the exact area of $R$.

(Solutions relying on calculator technology are not acceptable.)