| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2020 |
| Session | September |
| Marks | 7 |
| Topic | Parametric integration |
| Type | Show integral then evaluate area |
| Difficulty | Standard +0.3 This is a standard Further Maths parametric integration question requiring the formula A = ∫y(dx/dt)dt, followed by routine application of the double angle formula for sin²t. Both parts follow well-established techniques 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_5}
Figure 5 shows a sketch of the curve with parametric equations
$$x = 3\cos 2t, \quad y = 2\tan t, \quad 0 \leq t \leq \frac{\pi}{4}.$$
The region $R$, shown shaded in Figure 5, is bounded by the curve, the $x$-axis and the $y$-axis.
\begin{enumerate}[label=(\alph*)]
\item Show that the area of $R$ is given
$$\int_0^{\pi/4} 24\sin^2 t \, dt.$$ [4]
\item Hence, using algebraic integration, find the exact area of $R$. [3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2020 Q5 [7]}}