Parametric Equations Q2

The figure above shows an ellipse with parametric equations

\({x = 2\cos\theta, y = 6\sin\left(\theta + \frac{\pi}{3}\right), -\pi \leq \theta < \pi}\)

The curve meets the coordinate axes at points \({A}\), \({B}\), \({C}\) and \({D}\).

1. Find the coordinates of the points \(A\), \(B\), \(C\) and \(D\).

The straight line \(l\) is the tangent to the ellipse at the point \(E\).

2. State the equation of \(l\), given that \(l\) is parallel to the \(y\)-axis.
3. Find the value of \(\theta \) at the point \(E\).

The finite region bounded by the ellipse and the \(x\)-axis for which \(y\ge 0\) is shown shaded in the figure above.

4. Show that the area of this region is given by the integral
   \(\displaystyle\displaystyle \displaystyle\int_{-\frac{\pi }{3}}^{\frac{2\pi }{3}}{3-3\cos 2\theta +3\sqrt{3}\sin 2\theta \,\mathrm{d}\theta }\).
5. Hence find the area of the shaded region.