Emblem or applied region area

12. \begin{figure}[h] \end{figure} The curve shown in Figure 3 has parametric equations $$x = 6 \sin t \quad y = 5 \sin 2 t \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$ The region \(R\), shown shaded in Figure 3, is bounded by the curve and the \(x\)-axis. \begin{enumerate}[label=(\alph*)] \item

1. Show that the area of \(R\) is given by \(\int _ { 0 } ^ { \frac { \pi } { 2 } } 60 \sin t \cos ^ { 2 } t \mathrm {~d} t\)
2. Hence show, by algebraic integration, that the area of \(R\) is exactly 20 \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{e28350e9-5090-4079-97da-e669ef9a5a7a-34_451_570_1416_742} \captionsetup{labelformat=empty} \caption{Figure 4}
   \end{figure} Part of the curve is used to model the profile of a small dam, shown shaded in Figure 4. Using the model and given that

\includegraphics{figure_1} Figure 1 shows a cross-section \(R\) of a dam. The line \(AC\) is the vertical face of the dam, \(AB\) is the horizontal base and the curve \(BC\) is the profile. Taking \(x\) and \(y\) to be the horizontal and vertical axes, then \(A\), \(B\) and \(C\) have coordinates \((0, 0)\), \((3\pi^2, 0)\) and \((0, 30)\) respectively. The area of the cross-section is to be calculated. Initially the profile \(BC\) is approximated by a straight line.

1. Find an estimate for the area of the cross-section \(R\) using this approximation. [1]

The profile \(BC\) is actually described by the parametric equations. $$x = 16t^2 - \pi^2, \quad y = 30 \sin 2t, \quad \frac{\pi}{4} \leq t \leq \frac{\pi}{2}.$$

1. Find the exact area of the cross-section \(R\). [7]
2. Calculate the percentage error in the estimate of the area of the cross-section \(R\) that you found in part (a). [2]

A mathematics department is designing a new emblem to place on the walls outside its classrooms. The design for the emblem is shown in the diagram below. \includegraphics{figure_5} The emblem is modelled by the region between the \(x\)-axis and the curve with parametric equations $$x = 1 + 0.2t - \cos t, \quad y = k \sin^2 t,$$ where \(k\) is a positive constant and \(0 \leq t \leq \pi\). Lengths are in metres and the area of the emblem must be \(1 \text{m}^2\).

1. Show that \(k \int_0^\pi (0.2 + \sin t - 0.2 \cos^2 t - \sin t \cos^2 t) dt = 1\). [3]
2. Determine the exact value of \(k\). [6]