| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Error analysis for approximation |
| Difficulty | Moderate -0.3 This is a straightforward C2 integration question with standard trapezium rule application. Part (a) requires routine substitution into the trapezium formula, part (b) uses basic integration of x^{1/2}, and part (c) is simple percentage error calculation. While multi-part with 13 marks total, each component is textbook-standard with no conceptual challenges, making it slightly easier than average. |
| Spec | 1.08e Area between curve and x-axis: using definite integrals1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| \(x\) | 2 | 4 |
| \(1 + 3\sqrt{x}\) | 5.243 | 7 |
| area \(= \frac{1}{2} \times 2 \times [5.243 + 9.485 + 2(7 + 8.348)]\) | M1 A1 | |
| \(= 45.4\) (3sf) | B1 M1 A1 | |
| (b) \(= \int_2^8 (1 + 3\sqrt{x}) \, dx\) | ||
| \(= [x + 2x^{\frac{3}{2}}]_2^8\) | M1 A1 | |
| \(= [8 + 2(2\sqrt{2})^3] - [2 + 2(2\sqrt{2})]\) | M1 | |
| \(= (8 + 32\sqrt{2}) - (2 + 4\sqrt{2})\) | M1 | |
| \(= 6 + 28\sqrt{2}\) | A1 | |
| (c) \(= \frac{(6+28\sqrt{2})-45.4}{6+28\sqrt{2}} \times 100\% = 0.43\%\) | M1 A1 |
**(a)**
| $x$ | 2 | 4 | 6 | 8 |
|---|---|---|---|---|
| $1 + 3\sqrt{x}$ | 5.243 | 7 | 8.348 | 9.485 |
area $= \frac{1}{2} \times 2 \times [5.243 + 9.485 + 2(7 + 8.348)]$ | M1 A1 |
$= 45.4$ (3sf) | B1 M1 A1 |
**(b)** $= \int_2^8 (1 + 3\sqrt{x}) \, dx$ | |
$= [x + 2x^{\frac{3}{2}}]_2^8$ | M1 A1 |
$= [8 + 2(2\sqrt{2})^3] - [2 + 2(2\sqrt{2})]$ | M1 |
$= (8 + 32\sqrt{2}) - (2 + 4\sqrt{2})$ | M1 |
$= 6 + 28\sqrt{2}$ | A1 |
**(c)** $= \frac{(6+28\sqrt{2})-45.4}{6+28\sqrt{2}} \times 100\% = 0.43\%$ | M1 A1 | | (13 marks)
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**Total: 75 marks**The finite region $R$ is bounded by the curve $y = 1 + 3\sqrt{x}$, the $x$-axis and the lines $x = 2$ and $x = 8$.
\begin{enumerate}[label=(\alph*)]
\item Use the trapezium rule with three intervals of equal width to estimate to 3 significant figures the area of $R$. [6]
\item Use integration to find the exact area of $R$ in the form $a + b\sqrt{2}$. [5]
\item Find the percentage error in the estimate made in part (a). [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q9 [13]}}