8.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa7abe9f-f5c0-4578-afd1-73176c717536-16_607_983_255_541}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
Figure 2 shows a sketch of the curve with equation \(y = \sqrt { x } , x \geqslant 0\)
The region \(R\), shown shaded in Figure 2, is bounded by the curve, the line with equation \(x = 1\), the \(x\)-axis and the line with equation \(x = a\), where \(a\) is a constant.
Given that the area of \(R\) is 10
- find, in simplest form, the value of
- \(\int _ { 1 } ^ { a } \sqrt { 8 x } \mathrm {~d} x\)
- \(\int _ { 0 } ^ { a } \sqrt { x } \mathrm {~d} x\)
- show that \(a = 2 ^ { k }\), where \(k\) is a rational constant to be found.