Simpson's Rule Approximation

1. The diagram shows the curve with equation \(y = \ln ( 2 + \cos x ) , x \geq 0\).
The smallest value of \(x\) for which the curve meets the \(x\)-axis is \(a\) as shown.

1. Find the value of \(a\).
2. Use Simpson's rule with four strips of equal width to estimate the area of the region bounded by the curve in the interval \(0 \leq x \leq a\) and the coordinate axes.

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.

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 \(\int _ { a } ^ { b } f ( x ) d x\).

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.

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

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

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

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

2 Use Simpson's rule with 5 ordinates (4 strips) to find an approximation to $$\int _ { 1 } ^ { 3 } \frac { 1 } { \sqrt { 1 + x ^ { 3 } } } \mathrm {~d} x$$ giving your answer to three significant figures.

1 Use Simpson's rule with 5 ordinates (4 strips) to find an approximation to \(\int _ { 1 } ^ { 9 } \frac { 1 } { 1 + \sqrt { x } } \mathrm {~d} x\), giving your answer to three significant figures.