N2007 P1 Q11

A curve has parametric equations \(x={{\cos }^{2}}t\), \(y={{\sin }^{3}}t\), for \(0\le t\le \frac{\pi }{2}\).

1. Sketch the curve.[2]
2. The tangent to the curve at the point \(\left( {{\cos }^{2}}\theta ,\,\,{{\sin }^{3}}\theta \right)\) where \(0<\theta <\frac{\pi }{2}\), meets the \(x\)- and \(y\)-axes at \(Q\) and \(R\) respectively. The origin is denoted by \(O\). Show that the area of triangle \(OQR\) is \(\frac{1}{12}\sin \theta {{\left( 3{{\cos }^{2}}\theta +2{{\sin }^{2}}\theta \right)}^{2}}\).[6]
3. Show that the area under the curve for \(0\le t\le \frac{\pi }{2}\) is \(2 \int_{0}^{\frac{\pi }{2}}{\cos t{{\sin }^{4}}\mathrm{t}\,\,\mathrm{d}t}\), and use the substitution \(\sin t=u\) to find this area.[5]