10.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1f61f78b-5e77-4758-8ad5-ea00c7dfea2b-28_826_1632_264_153}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
The finite region \(R\), which is shown shaded in Figure 1, is bounded by the coordinate axes, the straight line \(l\) with equation \(y = \frac { 1 } { 3 } x + 5\) and the curve \(C\) with equation \(y = 4 x ^ { \frac { 1 } { 2 } } - x + 5 , x \geqslant 0\)
The line \(l\) meets the curve \(C\) at the point \(D\) on the \(y\)-axis and at the point \(E\), as shown in Figure 1.
- Use algebra to find the coordinates of the points \(D\) and \(E\).
The curve \(C\) crosses the \(x\)-axis at the point \(F\).
- Verify that the \(x\) coordinate of \(F\) is 25
- Use algebraic integration to find the exact area of the shaded region \(R\).