Trapezium Rule with Accuracy Analysis

2

1. Use the trapezium rule, with four strips each of width 0.5 , to estimate the value of $$\int _ { 0 } ^ { 2 } \mathrm { e } ^ { x ^ { 2 } } \mathrm {~d} x$$ giving your answer correct to 3 significant figures.
2. Explain how the trapezium rule could be used to obtain a more accurate estimate.

9 The trapezium rule is used to calculate 3 approximations to \(\int _ { 0 } ^ { 1 } \sqrt [ 3 ] { \sinh ( x ) } \mathrm { d } x\) with 1,2 and 4 strips respectively. The results are shown in Table 9.1. \begin{table}[h] \end{table}

1. Use these results to determine two approximations to \(\int _ { 0 } ^ { 1 } \sqrt [ 3 ] { \sinh ( x ) } \mathrm { d } x\) using Simpson's rule.
2. Use your answers to part (a) to state the value of \(\int _ { 0 } ^ { 1 } \sqrt [ 3 ] { \sinh ( x ) } \mathrm { d } x\) as accurately as you can, justifying the precision quoted. Table 9.2 shows some further approximations found using the trapezium rule, together with some analysis of these approximations. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Table 9.2}

   \(n\)	\(\mathrm { T } _ { n }\)	difference	ratio
   1	0.5276437
   2	0.6661765	0.138533
   4	0.7253427	0.059166	0.42709
   8	0.7498821	0.024539	0.41475
   16	0.7598858	0.010004	0.40766
   32	0.7639221	0.004036	0.40348
   64	0.7655404	0.001618	0.40095

   \end{table}
3. Explain what can be deduced about the order of the method in this case.
4. Use extrapolation to obtain the value of \(\int _ { 0 } ^ { 1 } \sqrt [ 3 ] { \sinh ( x ) } \mathrm { d } x\) as accurately as you can, justifying the precision quoted.

4

1. Use the trapezium rule with 1 strip to calculate an estimate of \(\int _ { 1 } ^ { 2 } \sqrt { 1 + x ^ { 3 } } \mathrm {~d} x\), giving your
   answer correct to six decimal places.
   [0pt] [2]
   Fig. 4 shows some spreadsheet output containing further approximations to this integral using the trapezium rule, denoted by \(T _ { n }\), and Simpson's rule, denoted by \(S _ { 2 n }\). \begin{table}[h]

   	A	B	C
   1	\(n\)	\(T _ { n }\)	\(S _ { 2 n }\)
   2	1		2.130135
   3	2	2.149378
   4	4	2.134751	2.129862
   5	8	2.131084

   \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{table}
2. Write down an efficient formula for cell C 4.
3. Find the value of \(S _ { 4 }\), giving your answer correct to 6 decimal places.
4. Without doing any further calculation, state the value of \(\int _ { 1 } ^ { 2 } \sqrt { 1 + x ^ { 3 } } \mathrm {~d} x\) as accurately as
   possible, justifying the precision quoted.
   [0pt] [2]
5. Use the fact that Simpson's rule is a fourth order method to obtain an improved approximation to the value of \(\int _ { 1 } ^ { 2 } \sqrt { 1 + x ^ { 3 } } \mathrm {~d} x\), stating the value of this integral to a precision which seems justified.

7 Sarah uses the trapezium rule to find a sequence of approximations to \(\int _ { 0 } ^ { 1 } \sqrt { \tanh ( x ) } \mathrm { d } x\).
Her spreadsheet output is shown in Fig. 7.1. \begin{table}[h] \end{table}

1. Write down the value of \(h\) used to find the approximation 0.62357601 .
2. Without doing any further calculation, state the value of \(\int _ { 0 } ^ { 1 } \sqrt { \tanh ( x ) } \mathrm { d } x\) as accurately as you
   can, justifying the precision quoted.
3. Explain what the values in the ratio column tell you about the order of convergence of this sequence of approximations. Sarah carries out further work using the midpoint rule and Simpson’s rule. Her results are shown in Fig. 7.2. \begin{table}[h]

   	M	N	O	P	Q	R
   1	\(n\)	\(T _ { n }\)	\(M _ { n }\)	\(S _ { 2 n }\)	difference	ratio
   2	1	0.43634681	0.679792	0.5986436
   3	2	0.5580694	0.64592745	0.61664144	0.018
   4	4	0.60199843	0.63374304	0.6231615	0.00652	0.362269
   5	8	0.61787073	0.62928129	0.62547777	0.00232	0.355253
   6	16	0.62357601	0.62765831	0.62629755	0.00082	0.35392
   7	32	0.62561716	0.62707259

   \captionsetup{labelformat=empty} \caption{Fig. 7.2}
   \end{table}
4. Write down an efficient spreadsheet formula for calculating \(S _ { 16 }\).
5. Determine the missing values in row 7.
6. Use extrapolation to determine the value of \(\int _ { 0 } ^ { 1 } \sqrt { \tanh ( x ) } d x\) as accurately as you can, justifying
   the precision quoted.
   [0pt] [6]

7 Fig. 7 shows the graph of \(y = \mathrm { f } ( x )\) for values of \(x\) from 0 to 1 . \begin{figure}[h] \end{figure} The following spreadsheet printout shows estimates of \(\int _ { 0 } ^ { 1 } f ( x ) d x\) found using the midpoint and trapezium rules for different values of \(h\), the strip width.

1. Without doing any further calculation, write down the smallest possible interval which contains the value of the integral. Justify your answer.
2. (A) - Calculate the ratio of differences, \(r\), for the sequence of estimates calculated using the trapezium rule.
   • Hence suggest a value for \(r\) correct to 2 significant figures.
   • Comment on your suggested value for \(r\).
     (B) - Use extrapolation to find an improved approximation to the value of the integral.
   • State the value of the integral to two decimal places.
   • Explain why this precision is secure.
   Using a similar approach with the sequence of estimates calculated using the midpoint rule, the approximation to the integral from extrapolation was found to be 1.94427 correct to 5 decimal places.
3. Andrea uses the extrapolated midpoint rule value and the value found in part (ii) ( \(B\) ) to write down an interval which contains the value of the integral. Comment on the validity of Andrea's method.
4. Use the values from the spreadsheet output to calculate an approximation to the integral using Simpson's rule with \(h = 0.125\). Give your answer to 5 decimal places. Approximations to the integral using Simpson's rule are given in the spreadsheet output below. The number of applications of Simpson's rule is given in column N.

   N	O	P	Q
   \(n\)	Simpson	differences	ratio
   1	1.91610623	0.01810005	0.3584931
   2	1.93420628	0.00648874	0.3556525
   4	1.94069502	0.00230774	0.3544828
   8	1.94300275	0.00081805	0.3539885
   16	1.94382081	0.00028958	0.3537638
   32	1.94411039	0.00010244	0.3536568
   64	1.94421283	\(3.623 \mathrm { E } - 05\)
   128	1.94424906

5. Use the spreadsheet output to find the value of the integral as accurately as possible. Justify the precision quoted. \section*{END OF QUESTION PAPER} OCR is committed to seeking permission to reproduce all third-party content that it uses in the assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
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2. (a) Use the trapezium rule, with four strips each of width 0.5 , to estimate the value of $$\int _ { 0 } ^ { 2 } \mathrm { e } ^ { x ^ { 2 } } \mathrm {~d} x$$ giving your answer correct to 3 significant figures.
(b) Explain how the trapezium rule could be used to obtain a more accurate estimate.
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