Use series to approximate integral

A question is this type if and only if it asks to use a series expansion to find an approximate value for a definite integral.

6 questions · Standard +0.5

4.08a Maclaurin series: find series for function
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CAIE Further Paper 2 2023 June Q1
6 marks Standard +0.8
1
  1. Find the Maclaurin series for \(\sin ^ { - 1 } x\) up to and including the term in \(x ^ { 3 }\).
  2. Deduce an approximation to \(\int _ { 0 } ^ { \frac { 1 } { 5 } } \frac { 1 } { \sqrt { 1 - u ^ { 2 } } } \mathrm {~d} u\), giving your answer as a fraction.
CAIE Further Paper 2 2024 June Q2
9 marks Challenging +1.2
2 The curve \(C\) has parametric equations $$x = \cosh t , \quad y = \sinh t , \quad \text { for } 0 < t \leqslant \frac { 3 } { 5 }$$ The length of \(C\) is denoted by \(s\).
  1. Show that \(s = \int _ { 0 } ^ { \frac { 3 } { 5 } } \sqrt { \cosh 2 t } \mathrm {~d} t\). \includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-04_2714_37_143_2008}
  2. By finding the Maclaurin's series for \(\sqrt { \cosh 2 t }\) up to and including the term in \(t ^ { 2 }\) ,deduce an approximation to \(s\) .
CAIE Further Paper 2 2020 November Q1
7 marks Standard +0.3
1
  1. By differentiating \(\mathrm { e } ^ { - x ^ { 2 } }\), find the Maclaurin's series for \(\mathrm { e } ^ { - x ^ { 2 } }\) up to and including the term in \(x ^ { 2 }\).
  2. Deduce an approximation to \(\int _ { 0 } ^ { \frac { 1 } { 5 } } \mathrm { e } ^ { - x ^ { 2 } } \mathrm {~d} x\), giving your answer as a rational fraction in its lowest terms.
OCR Further Pure Core 2 2021 June Q5
10 marks Standard +0.3
5 Let \(\mathrm { f } ( x ) = \sin ^ { - 1 } ( x )\).
    1. Determine \(f ^ { \prime \prime } ( x )\).
    2. Determine the first two non-zero terms of the Maclaurin expansion for \(\mathrm { f } ( x )\).
    3. By considering the first two non-zero terms of the Maclaurin expansion for \(\mathrm { f } ( \mathrm { x } )\), find an approximation to \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x\). Give your answer correct to 6 decimal places.
  2. By writing \(\mathrm { f } ( x )\) as \(\sin ^ { - 1 } ( x ) \times 1\), determine the value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x\). Give your answer in exact form.
CAIE Further Paper 2 2021 November Q1
5 marks Standard +0.3
It is given that \(y = \sinh(x^2) + \cosh(x^2)\).
  1. Use standard results from the list of formulae (MF19) to find the Maclaurin's series for \(y\) in terms of \(x\) up to and including the term in \(x^4\). [2]
  2. Deduce the value of \(\frac{d^4y}{dx^4}\) when \(x = 0\). [1]
  3. Use your answer to part (a) to find an approximation to \(\int_0^{\frac{1}{2}} y \, dx\), giving your answer as a rational fraction in its lowest terms. [2]
Edexcel CP1 2021 June Q2
7 marks Standard +0.3
  1. Use the Maclaurin series expansion for \(\cos x\) to determine the series expansion of \(\cos^2\left(\frac{x}{3}\right)\) in ascending powers of \(x\), up to and including the term in \(x^4\) Give each term in simplest form. [2]
  2. Use the answer to part (a) and calculus to find an approximation, to 5 decimal places, for $$\int_{\pi/6}^{\pi/4} \left(\frac{1}{x}\cos^2\left(\frac{x}{3}\right)\right)dx$$ [3]
  3. Use the integration function on your calculator to evaluate $$\int_{\pi/6}^{\pi/4} \left(\frac{1}{x}\cos^2\left(\frac{x}{3}\right)\right)dx$$ Give your answer to 5 decimal places. [1]
  4. Assuming that the calculator answer in part (c) is accurate to 5 decimal places, comment on the accuracy of the approximation found in part (b). [1]