1 The table below shows corresponding values of \(x\) and \(y\) for \(y = \sqrt { \frac { x } { 1 + x } }\)
The values of \(y\) are given to 4 significant figures.
| \(x\) | 0.5 | 1 | 1.5 | 2 | 2.5 |
| \(y\) | 0.5774 | 0.7071 | 0.7746 | 0.8165 | 0.8452 |
- Use the trapezium rule, with all the values of \(y\) in the table, to find an estimate for
$$\int _ { 0.5 } ^ { 2.5 } \sqrt { \frac { x } { 1 + x } } \mathrm {~d} x$$
giving your answer to 3 significant figures.
- Using your answer to part (a), deduce an estimate for \(\int _ { 0.5 } ^ { 2.5 } \sqrt { \frac { 9 x } { 1 + x } } \mathrm {~d} x\)
Given that
$$\int _ { 0.5 } ^ { 2.5 } \sqrt { \frac { 9 x } { 1 + x } } \mathrm {~d} x = 4.535 \text { to } 4 \text { significant figures }$$
- comment on the accuracy of your answer to part (b).