9 The constant \(a\) is such that \(\int _ { 0 } ^ { a } x \mathrm { e } ^ { \frac { 1 } { 2 } x } \mathrm {~d} x = 6\).
- Show that \(a\) satisfies the equation
$$x = 2 + \mathrm { e } ^ { - \frac { 1 } { 2 } x } .$$
- By sketching a suitable pair of graphs, show that this equation has only one root.
- Verify by calculation that this root lies between 2 and 2.5.
- Use an iterative formula based on the equation in part (i) to calculate the value of \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.