Area under parametric curve

\includegraphics{figure_2} Figure 2 shows a sketch of the curve defined by the parametric equations $$x = t^2 + 2t \quad y = \frac{2}{t(3-t)} \quad a \leq t \leq b$$ where \(a\) and \(b\) are constants. The ends of the curve lie on the line with equation \(y = 1\)

1. Find the value of \(a\) and the value of \(b\) [2]

The region \(R\), shown shaded in Figure 2, is bounded by the curve and the line with equation \(y = 1\)

1. Show that the area of region \(R\) is given by $$M - k \int_a^b \frac{t+1}{t(3-t)} dt$$ where \(M\) and \(k\) are constants to be found. [5]
2. 1. Write \(\frac{t+1}{t(3-t)}\) in partial fractions.
   2. Use algebraic integration to find the exact area of \(R\), giving your answer in simplest form. [6]

\includegraphics{figure_2} Part of the design of a stained glass window is shown in Fig. 2. The two loops enclose an area of blue glass. The remaining area within the rectangle \(ABCD\) is red glass. The loops are described by the curve with parametric equations $$x = 3 \cos t, \quad y = 9 \sin 2t, \quad 0 \leq t < 2\pi.$$

1. Find the cartesian equation of the curve in the form \(y^2 = f(x)\). [4]
2. Show that the shaded area in Fig. 2, enclosed by the curve and the \(x\)-axis, is given by $$\int_0^{\frac{\pi}{2}} A \sin 2t \sin t \, dt, \text{ stating the value of the constant } A.$$ [3]
3. Find the value of this integral. [4]

The sides of the rectangle \(ABCD\), in Fig. 2, are the tangents to the curve that are parallel to the coordinate axes. Given that 1 unit on each axis represents 1 cm,

1. find the total area of the red glass. [4]