- The curve \(C\) has equation
$$y = \frac { ( x - k ) ^ { 2 } } { \sqrt { x } } \quad x > 0$$
where \(k\) is a positive constant.
- Show that
$$\int _ { 1 } ^ { 16 } \frac { ( x - k ) ^ { 2 } } { \sqrt { x } } \mathrm {~d} x = a k ^ { 2 } + b k + \frac { 2046 } { 5 }$$
where \(a\) and \(b\) are integers to be found.
\begin{figure}[h]
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\caption{Figure 1}
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Figure 1 shows a sketch of the curve \(C\) and the line \(l\).
Given that \(l\) intersects \(C\) at the point \(A ( 1,9 )\) and at the point \(B ( 16 , q )\) where \(q\) is a constant, - show that \(k = 4\)
The region \(R\), shown shaded in Figure 1, is bounded by \(C\) and \(l\)
Using the answers to parts (a) and (b), - find the area of region \(R\)