| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2009 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Area under polynomial curve |
| Difficulty | Moderate -0.5 This is a straightforward C2 integration question requiring expansion of the quadratic, integration using standard power rule, and evaluation between given limits. The boundaries are provided, eliminating any problem-solving element, making it slightly easier than average but still requiring correct algebraic manipulation and integration technique. |
| Spec | 1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \(y=(1+x)(4-x)=4+3x-x^2\) | M1 | Expand, giving 3 or 4 terms |
| \(\int(4+3x-x^2)dx = 4x+\frac{3x^2}{2}-\frac{x^3}{3}\) | M1 A1 | M1: \(x^n \to x^{n+1}\) for at least one term; A1 correct, allow \(+c\) |
| \(=\left(16+24-\frac{64}{3}\right)-\left(-4+\frac{3}{2}+\frac{1}{3}\right)=\frac{125}{6}\) | M1 A1 | M1: substitute limits 4 and \(-1\) and subtract; A1 must be exact (\(=20\frac{5}{6}\)) |
| Total: [5] |
## Question 2:
| Working/Answer | Marks | Guidance |
|---|---|---|
| $y=(1+x)(4-x)=4+3x-x^2$ | M1 | Expand, giving 3 or 4 terms |
| $\int(4+3x-x^2)dx = 4x+\frac{3x^2}{2}-\frac{x^3}{3}$ | M1 A1 | M1: $x^n \to x^{n+1}$ for at least one term; A1 correct, allow $+c$ |
| $=\left(16+24-\frac{64}{3}\right)-\left(-4+\frac{3}{2}+\frac{1}{3}\right)=\frac{125}{6}$ | M1 A1 | M1: substitute limits 4 and $-1$ and subtract; A1 must be exact ($=20\frac{5}{6}$) |
| **Total: [5]** | | |
---\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{12e54724-64a3-4dc0-b7d5-6ef6cc04124c-03_870_1027_205_406}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows part of the curve $C$ with equation $y = ( 1 + x ) ( 4 - x )$.\\
The curve intersects the $x$-axis at $x = - 1$ and $x = 4$. The region $R$, shown shaded in Figure 1, is bounded by $C$ and the $x$-axis.
Use calculus to find the exact area of $R$.\\
\hfill \mbox{\textit{Edexcel C2 2009 Q2 [5]}}