N2018 P1 Q9

A curve \(C\) has parametric equations

\(x=2\theta -\sin 2\theta ,\) \(y=2{{\sin }^{2}}\theta \),

for \(0\le \theta \le \pi \).

1. Show that \(\frac{\mathrm{d}y}{\mathrm{d}x}=\cot \theta .\)[4]
2. The normal to the curve at the point where \(\theta =\alpha \) meets the \(x\)-axis at the point \(A\). Show that the \(x\)-coordinate of \(A\) is \(k\alpha \), where \(k\) is a constant to be found.[4]

Do not use a calculator in answering this part.

3. The distance between two points along a curve is the arc-length. Scientists and engineers need to use arc-length in applications such as finding the work done in moving an object along the path described by a curve or the length of the cabling used on a suspension bridge.
   The arc-length between two points on \(C\), where \(\theta =\beta \) and \(\theta =\gamma \), is given by the formula
   \(\displaystyle\displaystyle \displaystyle\int_{\beta }^{\gamma }{\sqrt{{{\left( \frac{\mathrm{d}x}{\mathrm{d}\theta } \right)}^{2}}+{{\left( \frac{\mathrm{d}y}{\mathrm{d}\theta } \right)}^{2}}}\mathrm{d}\theta }\)
   Find the total length of \(C\).[5]