7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3b73fe78-cc47-4615-9cfb-0b8d9ec0ffda-09_627_606_244_667}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
Figure 1 shows part of the curve with equation \(y = \sqrt { } ( \tan x )\). The finite region \(R\), which is bounded by the curve, the \(x\)-axis and the line \(x = \frac { \pi } { 4 }\), is shown shaded in Figure 1.
- Given that \(y = \sqrt { } ( \tan x )\), complete the table with the values of \(y\) corresponding to \(x = \frac { \pi } { 16 } , \frac { \pi } { 8 }\) and \(\frac { 3 \pi } { 16 }\), giving your answers to 5 decimal places.
| \(x\) | 0 | \(\frac { \pi } { 16 }\) | \(\frac { \pi } { 8 }\) | \(\frac { 3 \pi } { 16 }\) | \(\frac { \pi } { 4 }\) |
| \(y\) | 0 | | | | 1 |
- Use the trapezium rule with all the values of \(y\) in the completed table to obtain an estimate for the area of the shaded region \(R\), giving your answer to 4 decimal places.
The region \(R\) is rotated through \(2 \pi\) radians around the \(x\)-axis to generate a solid of revolution.
- Use integration to find an exact value for the volume of the solid generated.
\section*{LO}