Deduce related integral value

A question is this type if and only if it asks to use a trapezium rule result to deduce the value of a related integral through algebraic manipulation or transformation.

30 questions · Standard +0.0

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Edexcel Paper 2 2022 June Q5
6 marks Standard +0.3
  1. The table below shows corresponding values of \(x\) and \(y\) for \(y = \log _ { 3 } 2 x\) The values of \(y\) are given to 2 decimal places as appropriate.
\(x\)34.567.59
\(y\)1.6322.262.462.63
  1. Using the trapezium rule with all the values of \(y\) in the table, find an estimate for $$\int _ { 3 } ^ { 9 } \log _ { 3 } 2 x \mathrm {~d} x$$ Using your answer to part (a) and making your method clear, estimate
    1. \(\int _ { 3 } ^ { 9 } \log _ { 3 } ( 2 x ) ^ { 10 } \mathrm {~d} x\)
    2. \(\int _ { 3 } ^ { 9 } \log _ { 3 } 18 x \mathrm {~d} x\)
Edexcel Paper 2 2020 October Q1
5 marks Moderate -0.3
1 The table below shows corresponding values of \(x\) and \(y\) for \(y = \sqrt { \frac { x } { 1 + x } }\) The values of \(y\) are given to 4 significant figures.
\(x\)0.511.522.5
\(y\)0.57740.70710.77460.81650.8452
  1. Use the trapezium rule, with all the values of \(y\) in the table, to find an estimate for $$\int _ { 0.5 } ^ { 2.5 } \sqrt { \frac { x } { 1 + x } } \mathrm {~d} x$$ giving your answer to 3 significant figures.
  2. Using your answer to part (a), deduce an estimate for \(\int _ { 0.5 } ^ { 2.5 } \sqrt { \frac { 9 x } { 1 + x } } \mathrm {~d} x\) Given that $$\int _ { 0.5 } ^ { 2.5 } \sqrt { \frac { 9 x } { 1 + x } } \mathrm {~d} x = 4.535 \text { to } 4 \text { significant figures }$$
  3. comment on the accuracy of your answer to part (b).
Edexcel C2 Q5
9 marks Standard +0.3
  1. (a) Write down the exact value of \(\cos \frac { \pi } { 6 }\).
The finite region \(R\) is bounded by the curve \(y = \cos ^ { 2 } x\), where \(x\) is measured in radians, the positive coordinate axes and the line \(x = \frac { \pi } { 3 }\).
(b) Use the trapezium rule with three equally-spaced ordinates to estimate the area of \(R\), giving your answer to 3 significant figures. The finite region \(S\) is bounded by the curve \(y = \sin ^ { 2 } x\), where \(x\) is measured in radians, the positive coordinate axes and the line \(x = \frac { \pi } { 3 }\).
(c) Using your answer to part (b), find an estimate for the area of \(S\).
CAIE P2 2024 November Q6
9 marks Moderate -0.3
  1. Use the trapezium rule with two intervals to find an approximation to the area of region \(A\). Give your answer correct to 3 significant figures. \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-10_2720_38_105_2010} \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-11_2716_29_107_22}
  2. Find the exact total area of regions \(A\) and \(B\). Give your answer in the form \(k \ln m\), where \(k\) and \(m\) are constants.
  3. Deduce an approximation to the area of region \(B\). Give your answer correct to 3 significant figures.
  4. State, with a reason, whether your answer to part (c) is an over-estimate or an under-estimate of the area of region \(B\).
AQA Paper 1 2023 June Q5
4 marks Moderate -0.3
5
  1. Use the trapezium rule with 6 ordinates ( 5 strips) to find an approximate value for the shaded area. Give your answer to four decimal places.
    5
  2. Using your answer to part (a) deduce an estimate for \(\int _ { 1 } ^ { 4 } \frac { 20 } { \mathrm { e } ^ { x } - 1 } \mathrm {~d} x\)