1.09f Trapezium rule: numerical integration

1 Fig. 1 shows some spreadsheet output concerning the values of a function, \(\mathrm { f } ( x )\). \begin{table}[h] \end{table} The formula in cell B2 is ==EXP(-(A2\^{}2)) and equivalent formulae are in cells B3 to B6. The formula in cell C 2 is \(= \mathrm { B } 2\).
The formula in cell C3 is \(\quad = \mathrm { C } 2 + \mathrm { B } 3\). Equivalent formulae are in cells C4 to C6.

1. Use sigma notation to express the formula in cell C5 in standard mathematical notation.
2. Explain why the same value is displayed in cells C 5 and C 6. Now suppose that the value in cell C2 is chopped to 3 decimal places and used to approximate the value in cell C2.
3. Calculate the relative error when this approximation is used. Suppose that the values in cells B4, B5 and B6 are chopped to 3 decimal places and used as approximations to the original values in cells B4, B5 and B6 respectively.
4. Explain why the relative errors in these approximations are all the same.

5 Fig. 5 shows spreadsheet output concerning the estimation of the derivative of a function \(\mathrm { f } ( x )\) at \(x = 2\) using the forward difference method. \begin{table}[h] \end{table}

1. Write down suitable cell formulae for
   The formula in cell B2 is \(\quad = ( \mathrm { LN } ( \mathrm { SQRT } ( \mathrm { SINH } ( ( 2 + \mathrm { A } 2 ) \wedge 3 ) ) ) - \mathrm { LN } ( \mathrm { SQRT } ( \mathrm { SINH } ( 2 \wedge 3 ) ) ) ) / \mathrm { A } 2\) and equivalent formulae are entered in cells B3 to B17.
2. Write \(\mathrm { f } ( x )\) in standard mathematical notation. The value displayed in cell B17 is zero, even though the calculation results in a non-zero answer.
3. Explain how this has arisen.

6 The spreadsheet output in Fig. 6 shows approximations to \(\int _ { 0 } ^ { 1 } x ^ { - \sqrt { x } } \mathrm {~d} x\) found using the midpoint rule, denoted by \(M _ { n }\), and the trapezium rule, denoted by \(T _ { n }\). \begin{table}[h] \end{table}

1. Write down an efficient spreadsheet formula for cell C3.
2. By first completing the table in the Printed Answer Booklet using the Simpson's rule, calculate the most accurate estimate of \(\int _ { 0 } ^ { 1 } x ^ { - \sqrt { x } } \mathrm {~d} x\) that you can, justifying the precision quoted. \section*{END OF QUESTION PAPER}

2 The table shows some values of \(x\) and the associated values of \(y = f ( x )\).

1. Calculate an estimate of \(\frac { \mathrm { dy } } { \mathrm { dx } }\) at \(x = 3\) using the forward difference method, giving your answer correct to \(\mathbf { 5 }\) decimal places.
2. Calculate an estimate of \(\frac { \mathrm { dy } } { \mathrm { dx } }\) at \(x = 3\) using the central difference method, giving your answer correct to \(\mathbf { 5 }\) decimal places.
3. Explain why your answer to part (b) is likely to be closer than your answer to part (a) to the true value of \(\frac { \mathrm { dy } } { \mathrm { dx } }\) at \(x = 3\). When \(x = 5\) it is given that \(y = 1.4645\) and \(\frac { \mathrm { dy } } { \mathrm { dx } } = 0.1820\), correct to 4 decimal places.
4. Determine an estimate of the error when \(\mathrm { f } ( 5 )\) is used to estimate \(\mathrm { f } ( 5.024 )\).

5 Kai uses the midpoint rule, trapezium rule and Simpson's rule to find approximations to \(\int _ { \mathrm { a } } ^ { \mathrm { b } } \mathrm { f } ( \mathrm { x } ) \mathrm { dx }\), where \(a\) and \(b\) are constants. The associated spreadsheet output is shown in the table. Some of the values are missing.

1. Write down a suitable spreadsheet formula for cell H 5 .
2. Complete the copy of the table in the Printed Answer Booklet, giving the values correct to 7 decimal places.
3. Use your answers to part (b) to determine the value of \(\int _ { a } ^ { b } f ( x ) d x\) as accurately as you can, justifying the precision quoted.

3 The diagram shows the graph of \(y = f ( x )\) for values of \(x\) from 1 to 3.5. \includegraphics[max width=\textwidth, alt={}, center]{4023e87c-34b1-4abd-9acc-ede5e4d68c7f-03_945_1248_312_244} The table shows some values of \(x\) and the associated values of \(y\).

1. Use the forward difference method to calculate an approximation to \(\frac { \mathrm { dy } } { \mathrm { dx } }\) at \(x = 2\).
2. Use the central difference method to calculate an approximation to \(\frac { \mathrm { dy } } { \mathrm { dx } }\) at \(x = 2\).
3. On the copy of the diagram in the Printed Answer Booklet, show how the central difference method gives the approximation to \(\frac { \mathrm { dy } } { \mathrm { dx } }\) at \(x = 2\) which was found in part (b).
4. Explain whether your answer to part (a) or your answer to part (b) is likely to give a better approximation to \(\frac { \mathrm { dy } } { \mathrm { dx } }\) at \(x = 2\).

4 A spreadsheet is used to approximate \(\int _ { a } ^ { b } f ( x ) d x\) using the midpoint rule with 1 strip. The output is shown in the table below. The formula in cell C4 is \(= \mathrm { B } 4 \wedge ( 1 / \mathrm { B } 4 )\).
The formula in cell D4 is \(= 0.5 ^ { * } \mathrm { C } 4\).

1. Write the integral in standard mathematical notation. A graph of \(y = f ( x )\) is included in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{4023e87c-34b1-4abd-9acc-ede5e4d68c7f-04_789_1004_1199_235}
2. Explain whether 0.65518535 is an over-estimate or an under-estimate of \(\int _ { a } ^ { b } f ( x ) d x\).

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.

1 The table shows some values of \(x\), together with the associated values of a function, \(\mathrm { f } ( x )\).

1. Use the information in the table to calculate the most accurate estimate of \(f ^ { \prime } ( 2 )\) possible.
2. Calculate an estimate of the error when \(f ( 2 )\) is used as an estimate of \(f ( 2.05 )\).

6 Table 6.1 shows some values of \(x\) and the associated values of a function, \(y = f ( x )\). \begin{table}[h] \end{table}

1. Explain why it is not possible to use the central difference method to calculate an estimate of \(\frac { \mathrm { dy } } { \mathrm { dx } }\) when \(x = 1\).
2. Use the forward difference method to calculate an estimate of \(\frac { \mathrm { dy } } { \mathrm { dx } }\) when \(x = 1\). A student uses the forward difference method to calculate a series of approximations to \(\frac { \mathrm { dy } } { \mathrm { dx } }\) when \(x = 2\) with different values of the step length, \(h\). These approximations are shown in Table 6.2, together with some further analysis. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Table 6.2}

   \(h\)	0.8	0.4	0.2	0.1	0.05	0.025	0.0125	0.00625
   approximation	0.130452	0.138647	0.143381	0.145942	0.147277	0.147959	0.148304	0.148477
   difference		0.008195	0.004734	0.002561	0.001335	0.000682	0.000345	0.000173
   ratio			0.577633	0.541099	0.521186	0.510762	0.505424	0.502723

   \end{table}
3. 1. Explain what the ratios of differences tell you about the order of the method in this case.
   2. Comment on whether this is unusual.
4. Determine the value of \(\frac { \mathrm { dy } } { \mathrm { dx } }\) when \(x = 2\) as accurately as possible. You must justify the precision quoted.

7 A student is using a spreadsheet to find approximations to \(\int _ { 0 } ^ { 1 } f ( x ) d x\) using the midpoint rule, the trapezium rule and Simpson's rule. Some of the associated spreadsheet output with \(n = 1\) and \(n = 2\), is shown in Table 7.1. \begin{table}[h] \end{table}

1. Complete the copy of Table 7.1 in the Printed Answer Booklet. Give your answers correct to 5 decimal places.
2. State the value of \(\int _ { 0 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x\) as accurately as possible. You must justify the precision quoted. The student calculates some more approximations using Simpson's rule. These approximations are shown in the associated spreadsheet output, together with some further analysis, in Table 7.2. The values of \(S _ { 2 }\) and \(S _ { 4 }\) have been blacked out, together with the associated difference and ratio. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Table 7.2}

   n	\(\mathrm { S } _ { 2 n }\)	difference	ratio
   1
   2
   4	0.674353	-0.0209
   8	0.665199	-0.00915	0.438059
   16	0.661297	-0.0039	0.426286
   32	0.659675	-0.00162	0.415762
   64	0.659015	-0.00066	0.406785

   \end{table}
3. The student checks some of her values with a calculator. She does not obtain 0.406785 when she calculates \(- 0.00066 \div ( - 0.00162 )\). Explain whether the value in the spreadsheet, or her value, is a more precise approximation to the ratio of differences in this case.
4. 1. State the order of convergence of the values in the ratio column. You must justify your answer.
   2. Explain what the values in the ratio column tell you about the order of the method in this case.
   3. Comment on whether this is unusual.
5. Determine the value of \(\int _ { 0 } ^ { 1 } f ( x ) d x\) as accurately as you can. You must justify 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 Fig. 7.1 shows two values of \(x\) and the associated values of \(\mathrm { f } ( x )\). \begin{table}[h] \end{table}

1. Use the forward difference method to calculate an estimate of the gradient of \(\mathrm { f } ( x )\) at \(x = 3\), giving your answer correct to 4 decimal places. Fig. 7.2 shows some spreadsheet output with additional values of \(x\) and the associated values of \(\mathrm { f } ( x )\). \begin{table}[h]

   \(x\)	3	3.00001	3.0001	3.001	3.01	3.1
   \(\mathrm { f } ( x )\)	6.082763	6.08274	6.082541	6.08054	6.060454	5.848846

   \captionsetup{labelformat=empty} \caption{Fig. 7.2}
   \end{table} These values have been used to produce a sequence of estimates of the gradient of \(\mathrm { f } ( x )\) at \(x = 3\), together with some further analysis. This is shown in the spreadsheet output in Fig. 7.3. \begin{table}[h]

   \(h\)	0.1	0.01	0.001	0.0001	0.00001
   estimate	-2.339165	-2.230883	-2.220532	-2.219501	-2.219398
   difference	0.1082815	0.010352	0.0010307	0.000103
   ratio	0.095602	0.099567	0.0999568

   \captionsetup{labelformat=empty} \caption{Fig. 7.3}
   \end{table} Tommy states that the differences between successive estimates is decreasing so rapidly that the order of convergence of this sequence of estimates is much faster than first order.
2. Explain whether or not Tommy is correct.
3. Use extrapolation to determine the value of the gradient of \(\mathrm { f } ( x )\) at \(x = 3\) as accurately as possible, justifying the precision quoted.
4. Calculate an estimate of the absolute error when \(\mathrm { f } ( 3 )\) is used as an approximation to \(\mathrm { f } ( 3.02 )\).

2 The table shows some values of \(x\) and the associated values of \(\mathrm { f } ( x )\).

1. Complete the difference table in the Printed Answer Booklet.
2. Explain why the data may be interpolated by a polynomial of degree 2.
3. Use Newton's forward difference interpolation formula to obtain a polynomial of degree 2 for the data.

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]

5 A vehicle is moving in a straight line. Its velocity at different times is recorded and shown below. The velocities are recorded to 5 significant figures and the times may be assumed to be exact. It is suggested initially that a quadratic model may be appropriate for this situation.

1. Given that the vehicle is modelled as a particle with constant mass, what assumption about the net force acting on the vehicle leads to a quadratic model?
2. Find Newton's interpolating polynomial of degree 2 to model this situation. Write your answer in the form \(v = a t ^ { 2 } + b t + c\).
3. Comment on whether this model appears to be appropriate.
4. Use this model to find an approximation to the distance travelled over the interval \(5 \leq t \leq 15\). Further investigation suggests that a cubic model may be more appropriate.
5. What technique would you use to fit a cubic model to the data in the table?

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.
   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} }{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.
   OCR is part of the }\section*{}

1. Use Simpson's rule with 4 intervals to estimate

$$\int _ { 0.4 } ^ { 2 } e ^ { x ^ { 2 } } d x$$

2. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the vertical cross-section of the entrance to a tunnel. The width at the base of the tunnel entrance is 2 metres and its maximum height is 3 metres. The shape of the cross-section can be modelled by the curve with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = 3 \cos \left( \frac { \pi } { 2 } x ^ { 2 } \right) \quad x \in [ - 1,1 ]$$ A wooden door of uniform thickness 85 mm is to be made to seal the tunnel entrance.
Use Simpson's rule with 6 intervals to estimate the volume of wood required for this door, giving your answer in \(\mathrm { m } ^ { 3 }\) to 4 significant figures.

1. Use Simpson's Rule with 6 intervals to estimate

$$\int _ { 1 } ^ { 4 } \sqrt { 1 + x ^ { 3 } } d x$$