2.
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\caption{Figure 1}
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Figure 1 shows a sketch of part of the curve with equation \(y = x \mathrm { e } ^ { - \frac { 1 } { 2 } x } , x \geqslant 0\).
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 \(y = x e ^ { - \frac { 1 } { 2 } x }\).
| \(x\) | 0 | 1 | 2 | 3 | 4 |
| \(y\) | 0 | \(\mathrm { e } ^ { - \frac { 1 } { 2 } }\) | | \(3 \mathrm { e } ^ { - \frac { 3 } { 2 } }\) | \(4 \mathrm { e } ^ { - 2 }\) |
- Complete the table with the value of \(y\) corresponding to \(x = 2\)
- 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.
- Find \(\int x \mathrm { e } ^ { - \frac { 1 } { 2 } x } \mathrm {~d} x\).
- Hence find the exact area of \(R\), giving your answer in the form \(a + b \mathrm { e } ^ { - 2 }\), where \(a\) and \(b\) are integers.