7.
$$I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ^ { n } x \mathrm {~d} x , \quad n \geqslant 0$$
- Prove that, for \(n \geqslant 2\),
$$n I _ { n } = ( n - 1 ) I _ { n - 2 }$$
- \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1c262813-4160-4eda-9a36-e4ba38182c8a-22_588_1018_630_520}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
A designer is asked to produce a poster to completely cover the curved surface area of a solid cylinder which has diameter 1 m and height 0.7 m .
He uses a large sheet of paper with height 0.7 m and width of \(\pi \mathrm { m }\).
Figure 2 shows the first stage of the design, where the poster is divided into two sections by a curve.
The curve is given by the equation
$$y = \sin ^ { 2 } ( 4 x ) - \sin ^ { 10 } ( 4 x )$$
relative to axes taken along the bottom and left hand edge of the paper.
The region of the poster below the curve is shaded and the region above the curve remains unshaded, as shown in Figure 2.
Find the exact area of the poster which is shaded.