8.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{29b56d51-120a-4275-a761-8b8aed7bca54-24_560_1029_219_463}
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\caption{Figure 1}
\end{figure}
Figure 1 shows a sketch of part of the curve with equation \(y = \sqrt { \frac { x } { x ^ { 2 } + 1 } } , \quad x \geqslant 0\)
The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the line with equation \(x = 2\), the \(x\)-axis and the line with equation \(x = 7\)
The table below shows corresponding values of \(x\) and \(y\) for \(y = \sqrt { \frac { x } { x ^ { 2 } + 1 } }\)
| \(x\) | 2 | 3 | 4 | 5 | 6 | 7 |
| \(y\) | 0.6325 | 0.5477 | 0.4851 | 0.4385 | 0.4027 | 0.3742 |
- Use the trapezium rule, with all the values of \(y\) in the table, to find an estimate for the area of \(R\), giving your answer to 3 decimal places.
The region \(R\) is rotated \(360 ^ { \circ }\) about the \(x\)-axis to form a solid of revolution.
- Use calculus to find the exact volume of the solid of revolution formed. Write your answer in its simplest form.
\includegraphics[max width=\textwidth, alt={}, center]{29b56d51-120a-4275-a761-8b8aed7bca54-24_2255_47_314_1979}