| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2020 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Area under polynomial curve |
| Difficulty | Moderate -0.8 This is a straightforward P2 integration question requiring students to find intersection points, set up a single integral (either in x or y), and evaluate. It's a standard textbook exercise with no conceptual challenges—just routine application of area under a curve formulas with basic algebraic manipulation. |
| Spec | 1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks |
|---|---|
| \(\int (2x^2 + 7) \, dx = \frac{2x^3}{3} + 7x\) or \(\int (10 - 2x^2) \, dx = 10x - \frac{2x^3}{3}\) | M1 |
| Achieves/uses a limit of 5 | B1 |
| Area \(= 17\sqrt{5} - \int_0^5 (2x^2 + 7) \, dx\) or Area \(= \int_0^5 (10 - 2x^2) \, dx\) | M1 |
| \(= 17\sqrt{5} - \frac{2}{3} \times 5\sqrt{5} - 7\sqrt{5}\) or \(= 10\sqrt{5} - \frac{2}{3} \times 5\sqrt{5}\) | M1 |
| \(= \frac{20\sqrt{5}}{3}\) | A1 |
$\int (2x^2 + 7) \, dx = \frac{2x^3}{3} + 7x$ or $\int (10 - 2x^2) \, dx = 10x - \frac{2x^3}{3}$ | M1
Achieves/uses a limit of 5 | B1
Area $= 17\sqrt{5} - \int_0^5 (2x^2 + 7) \, dx$ or Area $= \int_0^5 (10 - 2x^2) \, dx$ | M1
$= 17\sqrt{5} - \frac{2}{3} \times 5\sqrt{5} - 7\sqrt{5}$ or $= 10\sqrt{5} - \frac{2}{3} \times 5\sqrt{5}$ | M1
$= \frac{20\sqrt{5}}{3}$ | A1
(6 marks)
---4.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{08aac50c-7317-4510-927a-7f5f2e00f485-08_858_654_118_671}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows a sketch of the curve with equation
$$y = 2 x ^ { 2 } + 7 \quad x \geqslant 0$$
The finite region $R$, shown shaded in Figure 1, is bounded by the curve, the $y$-axis and the line with equation $y = 17$
Find the exact area of $R$.\\
\hfill \mbox{\textit{Edexcel P2 2020 Q4 [6]}}