5. The curve \(C\) has equation
$$y = x \sqrt { } \left( x ^ { 3 } + 1 \right) , \quad 0 \leqslant x \leqslant 2$$
- Complete the table below, giving the values of \(y\) to 3 decimal places at \(x = 1\) and \(x = 1.5\).
| \(x\) | 0 | 0.5 | 1 | 1.5 | 2 |
| \(y\) | 0 | 0.530 | | | 6 |
- Use the trapezium rule, with all the \(y\) values from your table, to find an approximation for the value of \(\int _ { 0 } ^ { 2 } x \sqrt { } \left( x ^ { 3 } + 1 \right) \mathrm { d } x\), giving your answer to 3 significant figures.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{22ebc302-765c-4734-b312-b286ccb20be9-06_1110_644_1119_648}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
Figure 2 shows the curve \(C\) with equation \(y = x \sqrt { } \left( x ^ { 3 } + 1 \right) , 0 \leqslant x \leqslant 2\), and the straight line segment \(l\), which joins the origin and the point \(( 2,6 )\). The finite region \(R\) is bounded by \(C\) and \(l\). - Use your answer to part (b) to find an approximation for the area of \(R\), giving your answer to 3 significant figures.
(3)
\section*{LU}