1.
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\caption{Figure 1}
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Figure 1 shows a sketch of the curve with equation \(y = \frac { x } { 1 + \sqrt { x } } , x \geqslant 0\)
The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the line with equation \(x = 1\), the \(x\)-axis and the line with equation \(x = 3\)
The table below shows corresponding values of \(x\) and \(y\) for \(y = \frac { x } { 1 + \sqrt { } x }\)
| \(x\) | 1 | 1.5 | 2 | 2.5 | 3 |
| \(y\) | 0.5 | 0.6742 | 0.8284 | 0.9686 | 1.0981 |
- Use the trapezium rule, with all the values of \(y\) in the table, to find an estimate for the area of \(R\), giving your answer to 3 decimal places.
- Explain how the trapezium rule can be used to give a better approximation for the area of \(R\).
- Giving your answer to 3 decimal places in each case, use your answer to part (a) to deduce an estimate for
- \(\int _ { 1 } ^ { 3 } \frac { 5 x } { 1 + \sqrt { x } } \mathrm {~d} x\)
- \(\int _ { 1 } ^ { 3 } \left( 6 + \frac { x } { 1 + \sqrt { x } } \right) \mathrm { d } x\)