Mean value using inverse trig integral

A question is this type if and only if it requires finding the mean value of a function over an interval where the integration involves inverse trigonometric or hyperbolic functions.

2 questions · Standard +0.8

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CAIE FP1 2013 November Q6
8 marks Challenging +1.2
6 [In this question you may use, without proof, the formula \(\int \sec x \mathrm {~d} x = \ln ( \sec x + \tan x ) + \operatorname { const }\).]
  1. Let \(y = \sec x\). Find the mean value of \(y\) with respect to \(x\) over the interval \(\frac { 1 } { 6 } \pi \leqslant x \leqslant \frac { 1 } { 3 } \pi\).
  2. The curve \(C\) has equation \(y = - \ln ( \cos x )\), for \(0 \leqslant x \leqslant \frac { 1 } { 3 } \pi\). Find the arc length of \(C\).
OCR MEI Further Pure Core Specimen Q12
13 marks Standard +0.3
12 In this question you must show detailed reasoning.
  1. Given that \(y = \arctan x\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 1 + x ^ { 2 } }\). Fig. 12 shows the curve \(y = \frac { 1 } { 1 + x ^ { 2 } }\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{09d39832-5519-463d-ac7a-5d406ffd7be0-5_444_1435_1371_332} \captionsetup{labelformat=empty} \caption{Fig. 12}
    \end{figure}
  2. Find, in exact form, the mean value of the function \(\mathrm { f } ( x ) = \frac { 1 } { 1 + x ^ { 2 } }\) for \(- 1 \leq x \leq 1\).
  3. The region bounded by the curve, the \(x\)-axis, and the lines \(x = 1\) and \(x = - 1\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Find, in exact form, the volume of the solid of revolution generated.