| Exam Board | Edexcel |
|---|---|
| Module | PMT Mocks (PMT Mocks) |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric integration |
| Type | Show integral then evaluate area |
| Difficulty | Standard +0.3 This is a standard parametric area question requiring the formula A = ∫y(dx/dt)dt with correct limits, followed by routine integration using substitution u = sin t. The setup requires finding when x = 2 (giving t = π/3) and recognizing sin 2t = 2sin t cos t, but these are straightforward steps. The integration itself is a standard textbook exercise, making this slightly easier than average overall. |
| 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 |
16.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{48f9a252-61a2-491d-94d0-8470aee96942-24_771_1484_248_429}
\captionsetup{labelformat=empty}
\caption{Figure 8}
\end{center}
\end{figure}
Figure 8 shows a sketch of the curve with parametric equations
$$x = 4 \cos t \quad y = 2 \sin 2 t \quad 0 \leq t \leq \frac { \pi } { 2 }$$
where $t$ is a parameter.\\
The finite region $R$ is enclosed by the curve $C$, the $x$-axis and the line $x = 2$, as shown in Figure 7.\\
a. Show that the area of $R$ is given by
$$\int _ { \frac { \pi } { 3 } } ^ { \frac { \pi } { 2 } } 16 \sin ^ { 2 } t \cos t \mathrm {~d} t$$
b. Hence, using algebraic integration, find the exact area of $R$, giving in the form $a + b \sqrt { 3 }$, where $a$ and $b$ are constants to be determined.
\hfill \mbox{\textit{Edexcel PMT Mocks Q16 [6]}}