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\includegraphics[max width=\textwidth, alt={}, center]{1662cb34-273c-461d-908c-9fe2ffe889b4-06_599_1086_274_518} The diagram shows the curve with equation \(y = 10 x ^ { \frac { 1 } { 2 } } - \frac { 5 } { 2 } x ^ { \frac { 3 } { 2 } }\) for \(x > 0\). The curve meets the \(x\)-axis at the points \(( 0,0 )\) and \(( 4,0 )\). Find the area of the shaded region.
The diagram shows a sector \(O A B\) of a circle with centre \(O\). Angle \(A O B = \theta\) radians and \(O P = A P = x\).

1. Show that the arc length \(A B\) is \(2 x \theta \cos \theta\).
2. Find the area of the shaded region \(A P B\) in terms of \(x\) and \(\theta\).