3.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5c9f77f0-9f7c-4125-9da7-20fb8d79b05e-04_814_882_258_539}
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\caption{Figure 1}
\end{figure}
Figure 1 shows the finite region \(R\) bounded by the \(x\)-axis, the \(y\)-axis, the line \(x = \frac { \pi } { 2 }\) and the curve with equation
$$y = \sec \left( \frac { 1 } { 2 } x \right) , \quad 0 \leqslant x \leqslant \frac { \pi } { 2 }$$
The table shows corresponding values of \(x\) and \(y\) for \(y = \sec \left( \frac { 1 } { 2 } x \right)\).
| \(x\) | 0 | \(\frac { \pi } { 6 }\) | \(\frac { \pi } { 3 }\) | \(\frac { \pi } { 2 }\) |
| \(y\) | 1 | 1.035276 | | 1.414214 |
- Complete the table above giving the missing value of \(y\) to 6 decimal places.
- Using the trapezium rule, with all of the values of \(y\) from the completed table, find an approximation for the area of \(R\), giving your answer to 4 decimal places.
Region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
- Use calculus to find the exact volume of the solid formed.