\begin{exercice*}Dans chacun des cas suivants, écrire ou donner :
\begin{enumerate}
 \item l'expression de la fonction $f$ représentée en rouge ;
 \item la description du domaine coloré ;
 \item l'aire de ce domaine à l'aide d'une intégrale ;
 \item l'aire de ce domaine, en u.a.
 \begin{colenumerate}{2}
 \item \hfill\\
 \begin{tikzpicture}[general, xscale=0.6, yscale=0.6]
   \draw [quadrillage] (-3,-1) grid (2,4);
   \draw (-3,0)--(2,0);
   \draw (0,-1)--(0,4);
   \draw[color=black] (0,0) node[below left] {$O$};
   \draw[ultra thick, ->] (0,0)--(1,0);
   \draw[ultra thick, ->] (0,0)--(0,1);
   \draw[color=black] (0.5,0) node[below] {$\overrightarrow{i}$};
   \draw[color=black] (0,0.5) node[left] {$\overrightarrow{j}$};
   \draw[domain=-2:1, color=B2, smooth] plot(\x,{1.5});
   \draw (0.5,2.5) [color=B2] node{{\boldmath $\mathscr{C}_f$}};
   \fill[color=C1, opacity=0.2] (-2,0) -- (-2,1.5) --
 		plot[domain=-2:1] (\x,{1.5}) --
 		(1,0) -- cycle;
 \end{tikzpicture}
 
 \item \hfill\\
 \begin{tikzpicture}[general, xscale=0.6, yscale=0.6]
   \draw [quadrillage] (-2,-1) grid (3,4);
   \draw (-2,0)--(3,0);
   \draw (0,-1)--(0,4);
   \draw[color=black] (0,0) node[below left] {$O$};
   \draw[ultra thick, ->] (0,0)--(1,0);
   \draw[ultra thick, ->] (0,0)--(0,1);
   \draw[color=black] (0.5,0) node[below] {$\overrightarrow{i}$};
   \draw[color=black] (0,0.5) node[left] {$\overrightarrow{j}$};
   \draw[domain=-1:2, color=B2, smooth] plot(\x,{\x+1});
   \draw (0.5,2.5) [color=B2] node{{\boldmath $\mathscr{C}_f$}};
   \fill[color=C1, opacity=0.2] (-1,0) --
 		plot[domain=-1:2] (\x,{\x+1}) --
 		(2,0) -- cycle;
 \end{tikzpicture}
 
 \item \hfill\\
 \begin{tikzpicture}[general, xscale=0.6, yscale=1.2]
   \draw [quadrillage][xstep = 1, ystep = 0.5, line width = 0.1mm] (-2,-0.5) grid (3,2);
   \draw [quadrillage] (-2,-0.5) grid (3,2);
   \draw (-2,0)--(3,0);
   \draw (0,-0.5)--(0,2);
   \draw[color=black] (0,0) node[below left] {$O$};
   \draw[ultra thick, ->] (0,0)--(1,0);
   \draw[ultra thick, ->] (0,0)--(0,1);
   \draw[color=black] (0.5,0) node[below] {$\overrightarrow{i}$};
   \draw[color=black] (0,0.5) node[left] {$\overrightarrow{j}$};
   \draw[domain=-1:2, color=B2, smooth] plot(\x,{(\x+1)/2});
   \draw (0.5,1.25) [color=B2] node{{\boldmath $\mathscr{C}_f$}};
   \fill[color=C1, opacity=0.2] (-1,0) --
 		plot[domain=-1:2] (\x,{(\x+1)/2}) --
 		(2,0) -- cycle;
 \end{tikzpicture}
 
 \item \hfill\\
 \begin{tikzpicture}[general, xscale=0.6, yscale=0.6]
   \draw [quadrillage] (-3,-1) grid (2,4);
   \draw (-3,0)--(2,0);
   \draw (0,-1)--(0,4);
   \draw[color=black] (0,0) node[below left] {$O$};
   \draw[ultra thick, ->] (0,0)--(1,0);
   \draw[ultra thick, ->] (0,0)--(0,1);
   \draw[color=black] (0.5,0) node[below] {$\overrightarrow{i}$};
   \draw[color=black] (0,0.5) node[left] {$\overrightarrow{j}$};
   \draw[domain=-2:1.5, color=B2, smooth] plot(\x,{abs(\x)});
   \draw (0.5,1.5) [color=B2] node{{\boldmath $\mathscr{C}_f$}};
   \fill[color=C1, opacity=0.2] (-2,0) -- (-2,2) --
 		plot[domain=-2:1.5] (\x,{abs(\x)}) --
 		(1.5,0) -- cycle;
 \end{tikzpicture}
\end{colenumerate}
\end{enumerate}

\begin{corrige}
\begin{enumerate}
 \item $f(x) = 1,5$. $\mathscr{A} = \displaystyle{\int_{-2}^{1}}1,5 \, \mathrm{d}x = 4,5$ u.a.
 \item $f(x) = x+1$. $\mathscr{A} = \displaystyle{\int_{-1}^{2}}(x+1) \, \mathrm{d}x = 4,5$ u.a.
 \item $f(x) = \dfrac{x+1}{2}$. $\mathscr{A} = \displaystyle{\int_{-1}^{2}}\dfrac{x+1}{2} \, \mathrm{d}x = 2,25$ u.a.
 \item $f(x) = \left| x \right|$. $\mathscr{A} = \displaystyle{\int_{-2}^{1,5}}\left| x \right| \, \mathrm{d}x = 3,125$ u.a.
\end{enumerate}
\end{corrige}
\end{exercice*}

