\includegraphics{figure_1}

Figure 1 shows a sketch of part of the curve with equation $y = xe^{-\frac{1}{2}x}$, $x > 0$.

The finite region $R$, shown shaded in Figure 1, is bounded by the curve, the $x$-axis, and the line $x = 4$.

The table shows corresponding values of $x$ and $y$ for $y = xe^{-\frac{1}{2}x}$.

\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$x$ & 0 & 1 & 2 & 3 & 4 \\
\hline
$y$ & 0 & $e^{-\frac{1}{2}}$ & & $3e^{-\frac{3}{2}}$ & $4e^{-2}$ \\
\hline
\end{tabular}
\end{center}

\begin{enumerate}[label=(\alph*)]
\item Complete the table with the value of $y$ corresponding to $x = 2$
[1]

\item Use the trapezium rule, with all the values of $y$ in the completed table, to obtain an estimate for the area of $R$, giving your answer to 2 decimal places.
[4]

\item \begin{enumerate}[label=(\roman*)]
\item Find $\int xe^{-\frac{1}{2}x} \, dx$.

\item Hence find the exact area of $R$, giving your answer in the form $a + be^{-2}$, where $a$ and $b$ are integers.
[6]
\end{enumerate}
\end{enumerate}

\hfill \mbox{\textit{Edexcel C4 2013 Q2 [11]}}