- Solutions based entirely on graphical or numerical methods are not acceptable.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{aaf73eef-4103-48c2-865e-e8288891ae80-20_480_930_299_555}
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\caption{Figure 1}
\end{figure}
Figure 1 shows a sketch of part of the curve with equation
$$y = \operatorname { arsinh } x \quad x \geqslant 0$$
and the straight line with equation \(y = \beta\)
The line and the curve intersect at the point with coordinates \(( \alpha , \beta )\)
Given that \(\beta = \frac { 1 } { 2 } \ln 3\)
- show that \(\alpha = \frac { 1 } { \sqrt { 3 } }\)
The finite region \(R\), shown shaded in Figure 1, is bounded by the curve with equation \(y = \operatorname { arsinh } x\), the \(y\)-axis and the line with equation \(y = \beta\)
The region \(R\) is rotated through \(2 \pi\) radians about the \(y\)-axis.
- Use calculus to find the exact value of the volume of the solid generated.