1.09f Trapezium rule: numerical integration

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SPS SPS SM Pure 2023 September Q12
8 marks Standard +0.3
\includegraphics{figure_12} The figure above shows the curve \(C\) with equation $$f(x) = \frac{x+4}{\sqrt{x}}, \quad x > 0.$$
  1. Determine the coordinates of the minimum point of \(C\), labelled as \(M\). [5]
The point \(N\) lies on the \(x\) axis so that \(MN\) is parallel to the \(y\) axis. The finite region \(R\) is bounded by \(C\), the \(x\) axis, the straight line segment \(MN\) and the straight line with equation \(x = 1\).
  1. Use the trapezium rule with 4 strips of equal width to estimate the area of \(R\). [3]
OCR H240/03 2017 Specimen Q2
4 marks Easy -1.2
  1. Use the trapezium rule, with four strips each of width 0.25, to find an approximate value for \(\int_0^1 \frac{1}{\sqrt{1+x^2}} dx\). [3]
  2. Explain how the trapezium rule might be used to give a better approximation to the integral given in part (a). [1]
Pre-U Pre-U 9794/2 2010 June Q9
15 marks Challenging +1.2
  1. Show that $$\int x^n \ln x \, dx = \frac{x^{n+1}}{(n+1)^2}\left((n+1)\ln a - 1\right) + \frac{1}{(n+1)^2},$$ where \(n \neq -1\) and \(a > 1\). [6]
    1. Determine the \(x\)-coordinate of the point of intersection of the curves \(y = x^3 \ln x\) and \(y = x \ln 2^x\), where \(x > 0\). [2]
    2. Find the exact value of the area of the region enclosed between these two curves, the line \(x = 1\) and their point of intersection. Express your answer in the form \(b + c \ln 2\), where \(b\) and \(c\) are rational. [4]
  3. The curve \(y = (x^3 \ln x)^{0.5}\), for \(1 < x < e\), is rotated through \(2\pi\) radians about the \(x\)-axis. Determine the value of the resulting volume of revolution, giving your answer correct to 4 significant figures. [3]