Moderate -0.5 This is a straightforward application of the definition of a definite integral as a limit of Riemann sums. Part (a) requires recognizing the standard form and writing ∫₀.₂^1.8 (1/2x)dx. Part (b) involves routine integration of 1/x and evaluating limits, yielding ln(9). The question tests understanding of the fundamental concept but requires no problem-solving insight beyond direct application of the definition and basic integration rules.
4. a. Express \(\lim _ { \mathrm { d } x \rightarrow 0 } \sum _ { 0.2 } ^ { 1.8 } \frac { 1 } { 2 x } \delta x \quad\) as an integral.
b. Hence show that
$$\lim _ { \mathrm { d } x \rightarrow 0 } \sum _ { 0.2 } ^ { 1.8 } \frac { 1 } { 2 x } \delta x = \ln k$$
where \(k\) is a constant to be found.
\(\int_{0.2}^{1.8} \frac{1}{2x} \, dx\) or equivalent such as \(\frac{1}{2}\int_{0.2}^{1.8} x^{-1} \, dx\) but must include the limits and the \(dx\).
B1
(b)
\(\ln 3\)
A1
**(a)** | $\int_{0.2}^{1.8} \frac{1}{2x} \, dx$ or equivalent such as $\frac{1}{2}\int_{0.2}^{1.8} x^{-1} \, dx$ but must include the limits and the $dx$. | B1 | |
**(b)** | $\ln 3$ | A1 | M1 Know that $\int \frac{1}{x} \, dx = \ln x$ and attempts to apply the limits (either way round). Condone $\int \frac{1}{2x} \, dx = p \ln x$ (including $p=1$) or $\int \frac{1}{2x} \, dx = p \ln qx$ as long as the limits are applied. Also be aware that $\int \frac{1}{2x} \, dx = \ln x^2$, $\int \frac{1}{2x} \, dx = \frac{1}{2}\ln|x| + c$ and $\int \frac{1}{2x} \, dx = \frac{1}{2}\ln ax$ or equivalent are correct. $[p \ln x]_{0.2}^{1.8} = p \ln 1.8 - p \ln 0.2$ is sufficient evidence to award this mark. E.g. $\int_{0.2}^{1.8} \frac{1}{2x} \, dx = \left[\frac{1}{2}\ln|x|\right]_{0.2}^{1.8} = \frac{1}{2}[\ln 1.8 - \ln 0.2] = \frac{1}{2}\ln\frac{1.8}{0.2} = \frac{1}{2}\ln 9 = \ln 9^{\frac{1}{2}}$ |
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4. a. Express $\lim _ { \mathrm { d } x \rightarrow 0 } \sum _ { 0.2 } ^ { 1.8 } \frac { 1 } { 2 x } \delta x \quad$ as an integral.\\
b. Hence show that
$$\lim _ { \mathrm { d } x \rightarrow 0 } \sum _ { 0.2 } ^ { 1.8 } \frac { 1 } { 2 x } \delta x = \ln k$$
where $k$ is a constant to be found.\\
\hfill \mbox{\textit{Edexcel PMT Mocks Q4 [3]}}