| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Topic | Parametric curves and Cartesian conversion |
| Type | Area under parametric curve |
| Difficulty | Standard +0.3 This is a standard C4 parametric curves question with routine techniques: converting to Cartesian form using trigonometric identities, setting up an area integral using the parametric formula, evaluating using double-angle identities, and finding a rectangle area. All steps follow textbook methods with no novel insight required, making it slightly easier than average. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.08d Evaluate definite integrals: between limits1.08e Area between curve and x-axis: using definite integrals |
\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
$$x = 3 \cos t, \quad y = 9 \sin 2t, \quad 0 \leq t < 2\pi.$$
\begin{enumerate}[label=(\alph*)]
\item Find the cartesian equation of the curve in the form $y^2 = f(x)$. [4]
\item 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]
\item Find the value of this integral. [4]
\end{enumerate}
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,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item find the total area of the red glass. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 Q8 [15]}}