| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2015 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Area under curve |
| Difficulty | Moderate -0.3 This is a straightforward C2 integration question requiring expansion of brackets, application of standard power rule integration, and evaluation of a definite integral. The only mild complication is handling the two regions (one below, one above the x-axis), but the question explicitly guides students through this by asking for 'total area'. This is slightly easier than average due to its routine nature and clear structure, though not trivial as it requires multiple steps and careful handling of the absolute value for area. |
| Spec | 1.08b Integrate x^n: where n != -1 and sums1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Expands to give \(10x^{\frac{3}{2}} - 20x\) | B1 | Expands bracket correctly |
| Integrates to give \(\frac{10}{\frac{5}{2}}x^{\frac{5}{2}} + \frac{-20x^2}{2}\) \((+c)\) | M1 A1ft | Correct integration on at least one term; correct unsimplified follow through |
| Simplifies to \(4x^{\frac{5}{2}} - 10x^2\) \((+c)\) | A1cao | Must be simplified and correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use limits 0 and 4 on integrated function | M1 | May only see 4 substituted |
| Use limits 4 and 9 on integrated function | dM1 | Depends on first M mark |
| Obtains either \(\pm 32\) or \(\pm 194\) | A1 | Needs at least one previous M mark |
| Area \(= \left | \int_0^4 y\,dx\right | + \int_4^9 y\,dx = 32 + 194 = 226\) |
# Question 6:
## Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Expands to give $10x^{\frac{3}{2}} - 20x$ | B1 | Expands bracket correctly |
| Integrates to give $\frac{10}{\frac{5}{2}}x^{\frac{5}{2}} + \frac{-20x^2}{2}$ $(+c)$ | M1 A1ft | Correct integration on at least one term; correct unsimplified follow through |
| Simplifies to $4x^{\frac{5}{2}} - 10x^2$ $(+c)$ | A1cao | Must be simplified and correct |
## Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use limits 0 and 4 on integrated function | M1 | May only see 4 substituted |
| Use limits 4 and 9 on integrated function | dM1 | Depends on first M mark |
| Obtains either $\pm 32$ or $\pm 194$ | A1 | Needs at least one previous M mark |
| Area $= \left|\int_0^4 y\,dx\right| + \int_4^9 y\,dx = 32 + 194 = 226$ | ddM1, A1 | Adds 32 and 194; final answer 226 not $-226$ |
---6. (a) Find
$$\int 10 x \left( x ^ { \frac { 1 } { 2 } } - 2 \right) \mathrm { d } x$$
giving each term in its simplest form.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{8a7593c3-4f0b-4351-afae-7bd98cfc351d-10_401_1002_543_470}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
Figure 2 shows a sketch of part of the curve $C$ with equation
$$y = 10 x \left( x ^ { \frac { 1 } { 2 } } - 2 \right) , \quad x \geqslant 0$$
The curve $C$ starts at the origin and crosses the $x$-axis at the point $( 4,0 )$.
The area, shown shaded in Figure 2, consists of two finite regions and is bounded by the curve $C$, the $x$-axis and the line $x = 9$\\
(b) Use your answer from part (a) to find the total area of the shaded regions.
\hfill \mbox{\textit{Edexcel C2 2015 Q6 [9]}}