3 The polynomial \(\mathrm { p } ( x )\) is defined by $$p ( x ) = 6 x ^ { 3 } + a x ^ { 2 } + 3 x - 10$$ where \(a\) is a constant. It is given that \(( 2 x - 1 )\) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\) and hence factorise \(\mathrm { p } ( x )\) completely.
2. Solve the equation \(\mathrm { p } ( \operatorname { cosec } \theta ) = 0\) for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\).
   \includegraphics[max width=\textwidth, alt={}, center]{7b39a2ab-305d-43c5-a1e7-9442d6c13886-06_442_706_278_667} The diagram shows the curve with equation \(\mathrm { y } = \sqrt { 1 + \mathrm { e } ^ { 0.5 \mathrm { x } } }\). The shaded region is bounded by the curve and the straight lines \(x = 0 , x = 6\) and \(y = 0\).
3. Use the trapezium rule with three intervals to find an approximation to the area of the shaded region. Give your answer correct to 3 significant figures.
4. The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced.