Area under transcendental or composite curve

Find the exact area of a region bounded by a curve involving exponential, logarithmic, or trigonometric functions, the x-axis, and vertical lines, using direct integration.

2 questions · Standard +0.8

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OCR C3 Q8
11 marks Standard +0.3
  1. Given that \(y = \frac{4 \ln x - 3}{4 \ln x + 3}\), show that \(\frac{dy}{dx} = \frac{24}{x(4 \ln x + 3)^2}\). [3]
  2. Find the exact value of the gradient of the curve \(y = \frac{4 \ln x - 3}{4 \ln x + 3}\) at the point where it crosses the \(x\)-axis. [4]
  3. \includegraphics{figure_8iii} The diagram shows part of the curve with equation $$y = \frac{2}{x^2(4 \ln x + 3)}.$$ The region shaded in the diagram is bounded by the curve and the lines \(x = 1\), \(x = e\) and \(y = 0\). Find the exact volume of the solid produced when this shaded region is rotated completely about the \(x\)-axis. [4]
SPS SPS FM Pure 2026 November Q8
12 marks Challenging +1.3
In this question you must show detailed reasoning. The diagram shows the curve with equation \(y = \frac{x + 3}{\sqrt{x^2 + 9}}\). \includegraphics{figure_8} The region R, shown shaded in the diagram, is bounded by the curve, the \(x\)-axis, the \(y\)-axis, and the line \(x = 4\).
  1. Determine the area of R. Give your answer in the form \(p + \ln q\) where \(p\) and \(q\) are integers to be determined. [6]
The region R is rotated through \(2\pi\) radians about the \(x\)-axis.
  1. Determine the volume of the solid of revolution formed. Give your answer in the form \(\pi\left(a + b\ln\left(\frac{c}{d}\right)\right)\) where \(a\), \(b\), \(c\) and \(d\) are integers to be determined. [6]