4.
$$y = \frac { 5 } { \left( x ^ { 2 } + 1 \right) }$$
- Complete the table below, giving the missing value of \(y\) to 3 decimal places.
| \(x\) | 0 | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 |
| \(y\) | 5 | 4 | 2.5 | | 1 | 0.690 | 0.5 |
\includegraphics[alt={},max width=\textwidth]{1c51b071-5cb1-4841-b031-80bde9027433-06_732_1118_826_411}
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\caption{Figure 1}
\end{figure}
Figure 1 shows the region \(R\) which is bounded by the curve with equation \(y = \frac { 5 } { \left( x ^ { 2 } + 1 \right) }\),
the \(x\)-axis and the lines \(x = 0\) and \(x = 3\) the \(x\)-axis and the lines \(x = 0\) and \(x = 3\) - Use the trapezium rule, with all the values of \(y\) from your table, to find an approximate value for the area of \(R\).
- Use your answer to part (b) to find an approximate value for
$$\int _ { 0 } ^ { 3 } \left( 4 + \frac { 5 } { \left( x ^ { 2 } + 1 \right) } \right) d x$$
giving your answer to 2 decimal places.