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The diagram shows the curve with equation \(y = \sqrt { 2 x + 1 }\) between the points \(A \left( - \frac { 1 } { 2 } , 0 \right)\) and \(B ( 4,3 )\).
- Find the area of the region bounded by the curve, the \(x\)-axis and the line \(x = 4\). Hence find the area of the region bounded by the curve and the lines \(O A\) and \(O B\), where \(O\) is the origin.
- Show that the curve between \(B\) and \(A\) can be expressed in polar coordinates as
$$r = \frac { 1 } { 1 - \cos \theta } , \quad \text { where } \tan ^ { - 1 } \left( \frac { 3 } { 4 } \right) \leqslant \theta \leqslant \pi$$
- Deduce from parts (i) and (ii) that \(\int _ { \tan ^ { - 1 } \left( \frac { 3 } { 4 } \right) } ^ { \pi } \operatorname { cosec } ^ { 4 } \left( \frac { 1 } { 2 } \theta \right) \mathrm { d } \theta = 24\).
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