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.

3 questions · Challenging +1.4

4.08e Mean value of function: using integral
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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 \(C\).
WJEC Further Unit 4 2023 June Q8
11 marks Challenging +1.2
The function \(f\) is defined by $$f(x) = \frac{1}{\sqrt{x^2 + 4x + 3}}.$$
  1. Find the mean value of the function \(f\) for \(0 \leqslant x \leqslant 2\), giving your answer correct to three decimal places. [5]
  2. The region \(R\) is bounded by the curve \(y = f(x)\), the \(x\)-axis and the lines \(x = 0\) and \(x = 2\). Find the exact value of the volume of the solid generated when \(R\) is rotated through four right angles about the \(x\)-axis. [6]
WJEC Further Unit 4 2024 June Q7
12 marks Challenging +1.8
  1. A curve C is defined by the equation \(y = \frac{1}{\sqrt{16-6x-x^2}}\) for \(-3 \leq x \leq 1\).
    1. Find the mean value of \(y = \frac{1}{\sqrt{16-6x-x^2}}\) between \(x = -3\) and \(x = 1\). [4]
    2. The region \(R\) is bounded by the curve C, the \(x\)-axis and the lines \(x = -3\) and \(x = 1\). Find the volume of the solid generated when \(R\) is rotated through four right-angles about the \(x\)-axis. [5]
  2. Evaluate the improper integral $$\int_1^{\infty} \frac{8e^{-2x}}{4e^{-2x} - 5} \mathrm{d}x,$$ giving your answer correct to three decimal places. [3]