| Exam Board | Edexcel |
|---|---|
| Module | Paper 2 (Paper 2) |
| Year | 2019 |
| Session | June |
| Marks | 3 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Limit of sum as integral |
| Difficulty | Standard +0.3 This is a straightforward application of recognizing a Riemann sum as a definite integral. Students need only identify that the limit of the sum equals ∫₄⁹ √x dx and evaluate using the standard power rule for integration. The diagram provides significant scaffolding, making this easier than a typical integration question. |
| Spec | 1.08g Integration as limit of sum: Riemann sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| States \(\left\{\lim_{\delta x \to 0} \sum_{x=4}^{9} \sqrt{x}\ \delta x\ \text{is}\right\} \int_4^9 \sqrt{x}\ \mathrm{d}x\) | B1 | States \(\int_4^9 \sqrt{x}\ \mathrm{d}x\) with or without the '\(\mathrm{d}x\)' |
| \(= \left[\frac{2}{3}x^{\frac{3}{2}}\right]_4^9\) | M1 | Integrates \(\sqrt{x}\) to give \(\lambda x^{\frac{3}{2}},\ \lambda \neq 0\) |
| \(= \frac{2}{3} \times 9^{\frac{3}{2}} - \frac{2}{3} \times 4^{\frac{3}{2}} = \frac{54}{3} - \frac{16}{3}\) | ||
| \(= \frac{38}{3}\) or \(12\frac{2}{3}\) or awrt \(12.7\) | A1 |
## Question 5:
| Answer/Working | Mark | Guidance |
|---|---|---|
| States $\left\{\lim_{\delta x \to 0} \sum_{x=4}^{9} \sqrt{x}\ \delta x\ \text{is}\right\} \int_4^9 \sqrt{x}\ \mathrm{d}x$ | B1 | States $\int_4^9 \sqrt{x}\ \mathrm{d}x$ with or without the '$\mathrm{d}x$' |
| $= \left[\frac{2}{3}x^{\frac{3}{2}}\right]_4^9$ | M1 | Integrates $\sqrt{x}$ to give $\lambda x^{\frac{3}{2}},\ \lambda \neq 0$ |
| $= \frac{2}{3} \times 9^{\frac{3}{2}} - \frac{2}{3} \times 4^{\frac{3}{2}} = \frac{54}{3} - \frac{16}{3}$ | | |
| $= \frac{38}{3}$ or $12\frac{2}{3}$ or awrt $12.7$ | A1 | |
**Notes:**
- You can imply B1 for $\left[\lambda x^{\frac{3}{2}}\right]_4^9$ or for $\lambda \times 9^{\frac{3}{2}} - \lambda \times 4^{\frac{3}{2}}$
- Give B0 for $\int_1^9 \sqrt{x}\ \mathrm{d}x - \int_1^3 \sqrt{x}\ \mathrm{d}x$ or for $\int_3^9 \sqrt{x}\ \mathrm{d}x$ without reference to correct $\int_4^9 \sqrt{x}\ \mathrm{d}x$
- Give B1 M1 A1 for no working leading to $\frac{38}{3}$ or $12\frac{2}{3}$ or awrt 12.7
- Give B1 M1 A1 for $\left[\frac{2}{3}x^{\frac{3}{2}} + c\right]_4^9 = \frac{38}{3}$ or $12\frac{2}{3}$ or awrt 12.7
- Give M0 A0 for use of trapezium rule giving awrt 12.7, but allow B1 if $\int_4^9 \sqrt{x}\ \mathrm{d}x$ is seen in the method
- Otherwise give B0 M0 A0 for trapezium rule giving awrt 12.7
**(3 marks)**5.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{fa4afaf4-fe5d-4f3a-b3de-9600d5502a49-10_890_958_260_550}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
Figure 3 shows a sketch of the curve with equation $y = \sqrt { x }$\\
The point $P ( x , y )$ lies on the curve.\\
The rectangle, shown shaded on Figure 3, has height $y$ and width $\delta x$.\\
Calculate
$$\lim _ { \delta x \rightarrow 0 } \sum _ { x = 4 } ^ { 9 } \sqrt { x } \delta x$$
\hfill \mbox{\textit{Edexcel Paper 2 2019 Q5 [3]}}