We consider the function $f$ defined by $f(x) = \dfrac{x^2}{2}$.

We seek to determine the area $A$ of the surface delimited by the
graph of $f$, the $x$-axis and the lines of equations $x = 0$ and
$x = 5$. (see diagram below left)

\begin{center}
\includegraphics[height=7cm]{S7P3/2023_Bac_Reserve/2023_09_S7P3_FR_Bac_Reserve2_ExoA4_IMG1.jpg}
\qquad
\includegraphics[height=7cm]{S7P3/2023_Bac_Reserve/2023_09_S7P3_FR_Bac_Reserve2_ExoA4_IMG2.jpg}
\end{center}

\begin{enumerate}[a)]
    \item \nbPointsEnumi{2}Using the rectangles represented (see diagram above at
right), \textbf{determine} a lower and an upper bound of the area $A$ sought.

    \item \nbPointsEnumi{1}\textbf{Explain} how to obtain finer bounds with this
technique.

    \item \nbPointsEnumi{1}\textbf{Show} that the function $F$ defined by $F(x) = \dfrac{x^3}{6}$
    is an antiderivative of the function $f$ and calculate the exact value of the area $A$.
\end{enumerate}
