| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Session | Specimen |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Area under curve with fractional/negative powers or roots |
| Difficulty | Standard +0.3 This is a straightforward AS-level integration question requiring basic power rule integration of √x, simple algebraic manipulation, and solving for a constant. While it has multiple parts, each step follows standard procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.08b Integrate x^n: where n != -1 and sums1.08d Evaluate definite integrals: between limits1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(\int_1^a \sqrt{8x}\,dx = \sqrt{8}\times\int_1^a \sqrt{x}\,dx = 10\sqrt{8} = 20\sqrt{2}\) | M1 | Deduces \(\int_1^a\sqrt{8x}\,dx = \sqrt{8}\times\int_1^a\sqrt{x}\,dx\), attempting to multiply \(\int_1^a\sqrt{x}\,dx\) by \(\sqrt{8}\) |
| \(20\sqrt{2}\) | A1 | \(20\sqrt{2}\) or exact equivalent |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(\int_0^a \sqrt{x}\,dx = \int_0^1\sqrt{x}\,dx + \int_1^a\sqrt{x}\,dx = \left[\frac{2}{3}x^{\frac{3}{2}}\right]_0^1 + 10 = \frac{32}{3}\) | M1 | Identifies and attempts \(\int_0^a\sqrt{x}\,dx = \int_0^1\sqrt{x}\,dx + \int_1^a\sqrt{x}\,dx\) |
| \(\frac{32}{3}\) | A1 | \(\frac{32}{3}\) or exact equivalent |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(R = \int_1^a\sqrt{x}\,dx = \left[\frac{2}{3}x^{\frac{3}{2}}\right]_1^a\) | M1 A1 | Attempts to integrate \(x^{\frac{1}{2}}\rightarrow x^{\frac{3}{2}}\); correct integration |
| \(\frac{2}{3}a^{\frac{3}{2}}-\frac{2}{3}=10 \Rightarrow a^{\frac{3}{2}}=16 \Rightarrow a = 16^{\frac{2}{3}}\) | dM1 | Complete strategy to find \(a\); sets \(\left[\cdots x^{\frac{3}{2}}\right]_1^a = 10\) using both limits with correct index work |
| \(a = 2^{4\times\frac{2}{3}} = 2^{\frac{8}{3}}\) | A1 | Must see further statement following \(k=\frac{8}{3}\) if using \(a=2^k\) method |
## Question 8:
### Part (a)(i):
| Working | Mark | Guidance |
|---------|------|----------|
| $\int_1^a \sqrt{8x}\,dx = \sqrt{8}\times\int_1^a \sqrt{x}\,dx = 10\sqrt{8} = 20\sqrt{2}$ | M1 | Deduces $\int_1^a\sqrt{8x}\,dx = \sqrt{8}\times\int_1^a\sqrt{x}\,dx$, attempting to multiply $\int_1^a\sqrt{x}\,dx$ by $\sqrt{8}$ |
| $20\sqrt{2}$ | A1 | $20\sqrt{2}$ or exact equivalent |
### Part (a)(ii):
| Working | Mark | Guidance |
|---------|------|----------|
| $\int_0^a \sqrt{x}\,dx = \int_0^1\sqrt{x}\,dx + \int_1^a\sqrt{x}\,dx = \left[\frac{2}{3}x^{\frac{3}{2}}\right]_0^1 + 10 = \frac{32}{3}$ | M1 | Identifies and attempts $\int_0^a\sqrt{x}\,dx = \int_0^1\sqrt{x}\,dx + \int_1^a\sqrt{x}\,dx$ |
| $\frac{32}{3}$ | A1 | $\frac{32}{3}$ or exact equivalent |
### Part (b):
| Working | Mark | Guidance |
|---------|------|----------|
| $R = \int_1^a\sqrt{x}\,dx = \left[\frac{2}{3}x^{\frac{3}{2}}\right]_1^a$ | M1 A1 | Attempts to integrate $x^{\frac{1}{2}}\rightarrow x^{\frac{3}{2}}$; correct integration |
| $\frac{2}{3}a^{\frac{3}{2}}-\frac{2}{3}=10 \Rightarrow a^{\frac{3}{2}}=16 \Rightarrow a = 16^{\frac{2}{3}}$ | dM1 | Complete strategy to find $a$; sets $\left[\cdots x^{\frac{3}{2}}\right]_1^a = 10$ using both limits with correct index work |
| $a = 2^{4\times\frac{2}{3}} = 2^{\frac{8}{3}}$ | A1 | Must see further statement following $k=\frac{8}{3}$ if using $a=2^k$ method |
---8.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{fa7abe9f-f5c0-4578-afd1-73176c717536-16_607_983_255_541}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
Figure 2 shows a sketch of the curve with equation $y = \sqrt { x } , x \geqslant 0$\\
The region $R$, shown shaded in Figure 2, is bounded by the curve, the line with equation $x = 1$, the $x$-axis and the line with equation $x = a$, where $a$ is a constant.
Given that the area of $R$ is 10
\begin{enumerate}[label=(\alph*)]
\item find, in simplest form, the value of
\begin{enumerate}[label=(\roman*)]
\item $\int _ { 1 } ^ { a } \sqrt { 8 x } \mathrm {~d} x$
\item $\int _ { 0 } ^ { a } \sqrt { x } \mathrm {~d} x$
\end{enumerate}\item show that $a = 2 ^ { k }$, where $k$ is a rational constant to be found.
\end{enumerate}
\hfill \mbox{\textit{Edexcel AS Paper 1 Q8 [8]}}