The graph of $y = \frac{2x^3}{x^2 + 1}$ is shown for $0 \leq x \leq 4$

\begin{tikzpicture}[>=latex]

    % Define the mathematical function
    \def\myfunc{2*pow(\x,3) / (pow(\x,2) + 1)}

    % 1. Fill the shaded region 'A'
    \fill[gray!35] (0,0) -- plot[domain=0:4, samples=100] (\x, {\myfunc}) -- (4,0) -- cycle;

    % 2. Draw the grid
    % Vertical grid lines (every 0.2 units = 1/5)
    \foreach \i in {0,1,...,25} {
        \draw[gray!50, very thin] (\i/5, 0) -- (\i/5, 9);
    }
    % Horizontal grid lines (every 0.2 units = 1/5)
    \foreach \j in {0,1,...,45} {
        \draw[gray!50, very thin] (0, \j/5) -- (5, \j/5);
    }

    % 3. Draw the function curve
    \draw[line width=1.2pt] plot[domain=0:4, samples=100] (\x, {\myfunc});

    % 4. Draw the coordinate axes
    \draw[line width=0.8pt, ->] (0,0) -- (5.5,0) node[below] {$x$};
    \draw[line width=0.8pt, ->] (0,0) -- (0,9.6) node[left] {$y$};

    % 5. Ticks and Labels
    \node[below left] at (0,0) {0};

    \foreach \x in {2,4} {
        \draw[line width=0.8pt] (\x, 0) -- (\x, -0.1) node[below] {\x};
    }

    \foreach \y in {2,4,6,8} {
        \draw[line width=0.8pt] (0, \y) -- (-0.1, \y) node[left] {\y};
    }

    % 6. Label 'A'
    \node at (2.5, 1.2) {$A$};

\end{tikzpicture}

Caroline is attempting to approximate the shaded area, A, under the curve using the trapezium rule by splitting the area into $n$ trapezia.

\begin{enumerate}[label=(\alph*)]
\item When $n = 4$
\begin{enumerate}[label=(\roman*)]
\item State the number of ordinates that Caroline uses. [1 mark]
\item Calculate the area that Caroline should obtain using this method.

Give your answer correct to two decimal places. [3 marks]
\end{enumerate}

\item Show that the exact area of $A$ is

$$16 - \ln 17$$

Fully justify your answer. [5 marks]

\item Explain what would happen to Caroline's answer to part (a)(ii) as $n \to \infty$ [1 mark]
\end{enumerate}

\hfill \mbox{\textit{AQA Paper 1 2019 Q14 [10]}}