Area with logarithmic functions

A question is this type if and only if the curve equation involves logarithmic functions (ln x or log x) and requires integration techniques specific to logarithms.

3 questions · Standard +0.4

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OCR PURE 2022 June Q6
12 marks Standard +0.8
6 \includegraphics[max width=\textwidth, alt={}, center]{a1f4ccbd-f5ed-437a-ae76-c4925ce86e25-05_538_531_264_246} The shape \(A B C\) shown in the diagram is a student's design for the sail of a small boat.
The curve \(A C\) has equation \(y = 2 \log _ { 2 } x\) and the curve \(B C\) has equation \(y = \log _ { 2 } \left( x - \frac { 3 } { 2 } \right) + 3\).
  1. State the \(x\)-coordinate of point \(A\).
  2. Determine the \(x\)-coordinate of point \(B\).
  3. By solving an equation involving logarithms, show that the \(x\)-coordinate of point \(C\) is 2 . It is given that, correct to 3 significant figures, the area of the sail is 0.656 units \(^ { 2 }\).
  4. Calculate by how much the area is over-estimated or under-estimated when the curved edges of the sail are modelled as straight lines.
OCR MEI Paper 1 2018 June Q10
8 marks Challenging +1.2
10 Fig. 10 shows the graph of \(y = ( k - x ) \ln x\) where \(k\) is a constant ( \(k > 1\) ). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{904025c9-6d68-4344-bd41-8c0fccfcf92f-06_454_1266_1564_395} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure} Find, in terms of \(k\), the area of the finite region between the curve and the \(x\)-axis.
OCR H240/02 2023 June Q3
3 marks Moderate -0.8
3 In this question you must show detailed reasoning. Find the exact area of the region enclosed by the curve \(y = \frac { 1 } { x + 2 }\), the two axes and the line \(x = 2.5\).