Iterative formula from integral equation

A question is this type if and only if it involves an integral equation equal to a constant, requires showing a rearranged form, and then uses an iterative formula to find the value of a parameter to a specified accuracy.

6 questions · Standard +0.9

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CAIE P3 2011 November Q5
8 marks Standard +0.8
5 It is given that \(\int _ { 1 } ^ { a } x \ln x \mathrm {~d} x = 22\), where \(a\) is a constant greater than 1 .
  1. Show that \(a = \sqrt { } \left( \frac { 87 } { 2 \ln a - 1 } \right)\).
  2. Use an iterative formula based on the equation in part (i) to find the value of \(a\) correct to 2 decimal places. Use an initial value of 6 and give the result of each iteration to 4 decimal places.
CAIE P3 2014 November Q6
9 marks Standard +0.8
6 It is given that \(\int _ { 1 } ^ { a } \ln ( 2 x ) \mathrm { d } x = 1\), where \(a > 1\).
  1. Show that \(a = \frac { 1 } { 2 } \exp \left( 1 + \frac { \ln 2 } { a } \right)\), where \(\exp ( x )\) denotes \(\mathrm { e } ^ { x }\).
  2. Use the iterative formula $$a _ { n + 1 } = \frac { 1 } { 2 } \exp \left( 1 + \frac { \ln 2 } { a _ { n } } \right)$$ to determine the value of \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2017 November Q9
10 marks Challenging +1.2
9 It is given that \(\int _ { 1 } ^ { a } x ^ { \frac { 1 } { 2 } } \ln x \mathrm {~d} x = 2\), where \(a > 1\).
  1. Show that \(a ^ { \frac { 3 } { 2 } } = \frac { 7 + 2 a ^ { \frac { 3 } { 2 } } } { 3 \ln a }\).
  2. Show by calculation that \(a\) lies between 2 and 4 .
  3. Use the iterative formula $$a _ { n + 1 } = \left( \frac { 7 + 2 a _ { n } ^ { \frac { 3 } { 2 } } } { 3 \ln a _ { n } } \right) ^ { \frac { 2 } { 3 } }$$ to determine \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
CAIE P3 2022 June Q10
10 marks Standard +0.8
10 The constant \(a\) is such that \(\int _ { 1 } ^ { a } x ^ { 2 } \ln x \mathrm {~d} x = 4\).
  1. Show that \(a = \left( \frac { 35 } { 3 \ln a - 1 } \right) ^ { \frac { 1 } { 3 } }\).
  2. Verify by calculation that \(a\) lies between 2.4 and 2.8.
  3. Use an iterative formula based on the equation in part (a) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
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CAIE P3 2021 November Q8
10 marks Standard +0.8
8 The constant \(a\) is such that \(\int _ { 1 } ^ { a } \frac { \ln x } { \sqrt { x } } \mathrm {~d} x = 6\).
  1. Show that \(a = \exp \left( \frac { 1 } { \sqrt { a } } + 2 \right)\). \(\left[ \exp ( x ) \right.\) is an alternative notation for \(\left. \mathrm { e } ^ { x } .\right]\)
  2. Verify by calculation that \(a\) lies between 9 and 11 .
  3. Use an iterative formula based on the equation in part (a) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
AQA C4 2013 June Q8
10 marks Standard +0.8
8
  1. \(\quad\) Find \(\int t \cos \left( \frac { \pi } { 4 } t \right) \mathrm { d } t\).
  2. The platform of a theme park ride oscillates vertically. For the first 75 seconds of the ride, $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { t \cos \left( \frac { \pi } { 4 } t \right) } { 32 x }$$ where \(x\) metres is the height of the platform above the ground after time \(t\) seconds.
    At \(t = 0\), the height of the platform above the ground is 4 metres.
    Find the height of the platform after 45 seconds, giving your answer to the nearest centimetre.
    (6 marks)