A factory produces mineral water in bottles. The shape of the bottle labels is bounded by the x-axis and the curve $\mathscr { C }$ with equation $y = a \cos x$ with $x \in \left[ - \frac { \pi } { 2 } ; \frac { \pi } { 2 } \right]$ and $a$ a strictly positive real number.\\
A disk located inside is intended to receive information given to buyers. We consider the disk with centre at point A with coordinates $\left( 0 ; \frac { a } { 2 } \right)$ and radius $\frac { a } { 2 }$. It is admitted that this disk is entirely below the curve $\mathscr { C }$ for values of $a$ less than 1.4.

\begin{enumerate}
  \item Justify that the area of the region between the x-axis, the lines with equations $x = - \frac { \pi } { 2 }$ and $x = \frac { \pi } { 2 }$, and the curve $\mathscr { C }$ equals $2 a$ square units.
  \item For aesthetic reasons, it is desired that the area of the disk equals the area of the shaded surface. What value should be given to the real number $a$ to satisfy this constraint?
\end{enumerate}