6 The diagram shows the curve with equation \(y = \left( \mathrm { e } ^ { 3 x } + 1 \right) ^ { \frac { 1 } { 2 } }\) for \(x \geqslant 0\).
\includegraphics[max width=\textwidth, alt={}, center]{6ce5aa0d-0a73-4bc4-aabc-314c0434e4f5-5_483_611_402_717}

1. Find the gradient of the curve \(y = \left( \mathrm { e } ^ { 3 x } + 1 \right) ^ { \frac { 1 } { 2 } }\) at the point where \(x = \ln 2\).
2. Use the mid-ordinate rule with four strips to find an estimate for \(\int _ { 0 } ^ { 2 } \left( \mathrm { e } ^ { 3 x } + 1 \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x\), giving your answer to three significant figures.
3. The shaded region \(R\) is bounded by the curve, the lines \(x = 0 , x = 2\) and the \(x\)-axis. Find the exact value of the volume of the solid generated when the region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.