By rotating the cross-section figure around the x-axis, we obtain a model of the light bulb. We decompose it into three parts. We recall that:
\begin{itemize}
  \item the volume of a cylinder is given by the formula $\pi r^2 h$ where $r$ is the radius of the base disk and $h$ is the height;
  \item the volume of a sphere of radius $r$ is given by the formula $\frac{4}{3}\pi r^3$.
\end{itemize}
We also admit that, for every real number $x$ in the interval $[0;4]$, $f(x) = 2 - \cos\left(\frac{\pi}{4}x\right)$.

The points are $\mathrm{A}(-1;1)$, $\mathrm{B}(0;1)$, $\mathrm{C}(4;3)$, $\mathrm{D}(7;0)$, $\mathrm{E}(4;-3)$, $\mathrm{F}(0;-1)$, $\mathrm{G}(-1;-1)$.

\begin{enumerate}
  \item Calculate the volume of the cylinder with cross-section the rectangle $ABFG$.
  \item Calculate the volume of the hemisphere with cross-section the half-disk with diameter $[CE]$.
  \item To approximate the volume of the solid with cross-section the shaded region BCEF, we divide the segment $[OO']$ into $n$ segments of equal length $\frac{4}{n}$ then we construct $n$ cylinders of equal height $\frac{4}{n}$.\\
  a. Special case: in this question only we choose $n = 5$. Calculate the volume of the third cylinder, shaded in the figures, then give its value rounded to $10^{-2}$.\\
  b. General case: in this question, $n$ denotes any non-zero natural number. We approximate the volume of the solid with cross-section BCEF by the sum of the volumes of the $n$ cylinders thus created by choosing a sufficiently large value of $n$. Copy and complete the following algorithm so that at the end of its execution, the variable $V$ contains the sum of the volumes of the $n$ cylinders created when $n$ is entered.
\begin{verbatim}
$V \leftarrow 0$
For $k$ going from...to ... :
    $\mid V \leftarrow \ldots$
Fin For
\end{verbatim}
\end{enumerate}