3 Fig. 4 shows the curve \(y = \sqrt { 1 + \mathrm { e } ^ { 2 x } }\), and the region between the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = 2\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ce44db53-2ec8-497b-a1d5-a8adf85e3929-3_656_736_482_665}
\captionsetup{labelformat=empty}
\caption{Fig. 4}
\end{figure}
- Find the exact volume of revolution when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
- Complete the table of values, and use the trapezium rule with 4 strips to estimate the area of the shaded region.
| \(x\) | 0 | 0.5 | 1 | 1.5 | 2 |
| \(y\) | | 1.9283 | 2.8964 | 4.5919 | |
- The trapezium rule for \(\int _ { 0 } ^ { 2 } \sqrt { 1 + \mathrm { e } ^ { 2 x } } \mathrm {~d} x\) with 8 and 16 strips gives 6.797 and 6.823, although not necessarily in that order. Without doing the calculations, say which result is which, explaining your reasoning.