| Exam Board | Edexcel |
|---|---|
| Module | PMT Mocks (PMT Mocks) |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Complete table then estimate |
| Difficulty | Standard +0.8 This question requires standard trapezium rule application (parts a-b) but part (c) demands recognition of a non-trivial substitution relationship between integrals. Students must identify that the integrand √(2^(2x) + 2x) relates to the original through u = 2x, requiring careful handling of limits and the factor of 2 from du/dx. This conceptual leap beyond routine numerical integration elevates it above average difficulty. |
| Spec | 1.09f Trapezium rule: numerical integration |
| \(x\) | 0 | 0.2 | 0.4 | 0.6 | 0.8 | 1 |
| \(y\) | 1 | 1.161 | 1.311 | 1.732 |
| Answer | Marks | Guidance |
|---|---|---|
| Complete table with y values to 3 decimal places | (1) | B1 for each correct value |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int_0^1 \sqrt{2^x + x} \, dx \approx 1.38\) (to 3 significant figures) | (3) | B1 Uses strip width of 0.2 units; M1 Uses correct form of trapezium bracket; A1 Answer 1.38 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int_0^{0.5} \sqrt{2^{2x} + 2x} \, dx = 0.69\) | (2) | M1 Attempts to use answer to (b) × \(\frac{1}{2}\); A1 0.69 |
**Part (a):**
Complete table with y values to 3 decimal places | (1) | B1 for each correct value
**Part (b):**
$\int_0^1 \sqrt{2^x + x} \, dx \approx 1.38$ (to 3 significant figures) | (3) | B1 Uses strip width of 0.2 units; M1 Uses correct form of trapezium bracket; A1 Answer 1.38
**Part (c):**
$\int_0^{0.5} \sqrt{2^{2x} + 2x} \, dx = 0.69$ | (2) | M1 Attempts to use answer to (b) × $\frac{1}{2}$; A1 0.69
---1.
$$y = \sqrt { \left( 2 ^ { x } + x \right) }$$
a. Complete the table below, giving the values of $y$ to 3 decimal places.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | }
\hline
$x$ & 0 & 0.2 & 0.4 & 0.6 & 0.8 & 1 \\
\hline
$y$ & 1 & 1.161 & 1.311 & & & 1.732 \\
\hline
\end{tabular}
\end{center}
(1)\\
b. Use the trapezium rule with all the values of $y$ from your table to find an approximation for the value of
$$\int _ { 0 } ^ { 1 } \sqrt { \left( 2 ^ { x } + x \right) } \mathrm { d } x$$
giving your answer to 3 significant figures.
Using your answer to part (b) and making your method clear, estimate\\
c. $\int _ { 0 } ^ { 0.5 } \sqrt { \left( 2 ^ { 2 x } + 2 x \right) } \mathrm { d } x$\\
\hfill \mbox{\textit{Edexcel PMT Mocks Q1 [6]}}