8 The constant \(a\) is such that \(\int _ { 1 } ^ { a } \frac { \ln x } { \sqrt { x } } \mathrm {~d} x = 6\).
- Show that \(a = \exp \left( \frac { 1 } { \sqrt { a } } + 2 \right)\).
\(\left[ \exp ( x ) \right.\) is an alternative notation for \(\left. \mathrm { e } ^ { x } .\right]\) - Verify by calculation that \(a\) lies between 9 and 11 .
- Use an iterative formula based on the equation in part (a) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.