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\includegraphics[max width=\textwidth, alt={}, center]{8b051aee-4920-42a0-8b74-cbfa9f3c1ab1-3_373_623_260_760} The diagram shows the curve \(y = \sqrt { } \left( 1 + \mathrm { e } ^ { \frac { 1 } { 3 } x } \right)\) for \(0 \leqslant x \leqslant 6\). The region bounded by the curve and the lines \(x = 0 , x = 6\) and \(y = 0\) is denoted by \(R\).

1. Use the trapezium rule with 2 strips to find an estimate of the area of \(R\), giving your answer correct to 2 decimal places.
2. With reference to the diagram, explain why this estimate is greater than the exact area of \(R\).
3. The region \(R\) is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced.