2007 H2 Mathematics

Sketch the curve.

The tangent to the curve at the point $({\cos }^2\theta ,{\sin }^3\theta ),$ where $0<\theta <\frac{1}{2}\pi ,$ meets the $x\text{-}$ and $y\text{-axes}$ at $Q$ and $R$ respectively.

The origin is denoted by $O.$ Show that the area of triangle $OQR$ is

$$\frac{1}{12}\sin \theta {(3{\cos }^2\theta +2{\sin }^2\theta )}^2.$$

Show that the area under the curve for $0\le t\le \frac{1}{2}\pi$ is

$${\int }_0^{\frac{1}{2}\pi }\cos t{\sin }^{4}t\mathrm{d}\mathrm{t},$$

and use the substitution $\sin t=u$ to find this area.