10 Let $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { n } x \mathrm {~d} x$$ where \(n \geqslant 0\). Show that, for all \(n \geqslant 2\), $$I _ { n } = \frac { n - 1 } { n } I _ { n - 2 }$$ A curve has parametric equations \(x = a \sin ^ { 3 } t\) and \(y = a \cos ^ { 3 } t\), where \(a\) is a constant and \(0 \leqslant t \leqslant \frac { 1 } { 2 } \pi\). Show that the mean value \(m\) of \(y\) over the interval \(0 \leqslant x \leqslant a\) is given by $$m = 3 a \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \left( \cos ^ { 4 } t - \cos ^ { 6 } t \right) \mathrm { d } t$$ Find the exact value of \(m\), in terms of \(a\).

10 Let

$$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { n } x \mathrm {~d} x$$

where $n \geqslant 0$. Show that, for all $n \geqslant 2$,

$$I _ { n } = \frac { n - 1 } { n } I _ { n - 2 }$$

A curve has parametric equations $x = a \sin ^ { 3 } t$ and $y = a \cos ^ { 3 } t$, where $a$ is a constant and $0 \leqslant t \leqslant \frac { 1 } { 2 } \pi$. Show that the mean value $m$ of $y$ over the interval $0 \leqslant x \leqslant a$ is given by

$$m = 3 a \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \left( \cos ^ { 4 } t - \cos ^ { 6 } t \right) \mathrm { d } t$$

Find the exact value of $m$, in terms of $a$.

\hfill \mbox{\textit{CAIE FP1 2011 Q10}}