Simpson's rule application

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.

1. Use Simpson's rule with four strips to estimate the value of the integral

$$\int _ { 0 } ^ { 3 } \mathrm { e } ^ { \cos x } \mathrm {~d} x$$

4 The integral I is defined by $$I = \int _ { 0 } ^ { 13 } ( 2 x + 1 ) ^ { \frac { 1 } { 3 } } d x$$

1. Use integration to find the exact value of I .
2. Use Simpson's rule with two strips to find an approximate value for I. Give your answer correct to 3 significant figures.

7

1. Find the exact value of \(\int _ { 1 } ^ { 9 } ( 7 x + 1 ) ^ { \frac { 1 } { 3 } } \mathrm {~d} x\).
2. Use Simpson's rule with two strips to show that an approximate value of \(\int _ { 1 } ^ { 9 } ( 7 x + 1 ) ^ { \frac { 1 } { 3 } } \mathrm {~d} x\) can be expressed in the form \(m + n \sqrt [ 3 ] { 36 }\), where the values of the constants \(m\) and \(n\) are to be stated.
3. Use the results from parts (i) and (ii) to find an approximate value of \(\sqrt [ 3 ] { 36 }\), giving your answer in the form \(\frac { p } { q }\) where \(p\) and \(q\) are integers. \section*{Question 8 begins on page 4.}

1 Use Simpson's rule, with five ordinates (four strips), to calculate an estimate for $$\int _ { 0 } ^ { \pi } x ^ { \frac { 1 } { 2 } } \sin x d x$$ Give your answer to four significant figures.
[0pt] [4 marks]

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.

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.

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 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]

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

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

1. (a) Given that

$$y = \ln \left( 3 + x ^ { 2 } \right)$$ complete the table with the value of \(y\) corresponding to \(x = 3\), giving your answer to 4 significant figures. In part (b) you must show all stages of your working. \section*{Solutions relying entirely on calculator technology are not acceptable.} (b) Use Simpson's rule with all the values of \(y\) in the completed table to estimate, to 3 significant figures, the value of $$\int _ { 2 } ^ { 5 } \ln \left( 3 + x ^ { 2 } \right) \mathrm { d } x$$ (c) Using your answer to part (b) and making your method clear, estimate the value of $$\int _ { 2 } ^ { 5 } \ln \sqrt { \left( 3 + x ^ { 2 } \right) } \mathrm { d } x$$

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.

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.

5

1. Use the mid-ordinate rule with four strips to find an estimate for \(\int _ { 0 } ^ { 12 } \ln \left( x ^ { 2 } + 5 \right) \mathrm { d } x\), giving your answer to three significant figures.
2. A curve has equation \(y = \ln \left( x ^ { 2 } + 5 \right)\).
   1. Show that this equation can be rewritten as \(x ^ { 2 } = \mathrm { e } ^ { y } - 5\).
   2. The region bounded by the curve, the lines \(y = 5\) and \(y = 10\) and the \(y\)-axis is rotated through \(360 ^ { \circ }\) about the \(y\)-axis. Find the exact value of the volume of the solid generated.
3. The graph with equation \(y = \ln \left( x ^ { 2 } + 5 \right)\) is stretched with scale factor 4 parallel to the \(x\)-axis, and then translated through \(\left[ \begin{array} { l } 0 \\ 3 \end{array} \right]\) to give the graph with equation \(y = \mathrm { f } ( x )\). Write down an expression for \(\mathrm { f } ( x )\).

The diagram shows the curve with equation \(y = \ln(6x)\). \includegraphics{figure_1}

1. State the \(x\)-coordinate of the point of intersection of the curve with the \(x\)-axis. [1]
2. Find \(\frac{dy}{dx}\). [2]
3. Use Simpson's rule with 6 strips (7 ordinates) to find an estimate for \(\int_1^7 \ln(6x) \, dx\), giving your answer to three significant figures. [4]

\includegraphics{figure_8} The diagram shows part of the curve \(y = \ln(5 - x^2)\) which meets the \(x\)-axis at the point \(P\) with coordinates \((2, 0)\). The tangent to the curve at \(P\) meets the \(y\)-axis at the point \(Q\). The region \(A\) is bounded by the curve and the lines \(x = 0\) and \(y = 0\). The region \(B\) is bounded by the curve and the lines \(PQ\) and \(x = 0\).

1. Find the equation of the tangent to the curve at \(P\). [5]
2. Use Simpson's Rule with four strips to find an approximation to the area of the region \(A\), giving your answer correct to 3 significant figures. [4]
3. Deduce an approximation to the area of the region \(B\). [2]

\includegraphics{figure_8} The diagram shows the curve with equation \(y = x^8 e^{-x^2}\). The curve has maximum points at \(P\) and \(Q\). The shaded region \(A\) is bounded by the curve, the line \(y = 0\) and the line through \(Q\) parallel to the \(y\)-axis. The shaded region \(B\) is bounded by the curve, the line \(y = 0\) and the line \(PQ\).

1. Show by differentiation that the \(x\)-coordinate of \(Q\) is 2. [5]
2. Use Simpson's rule with 4 strips to find an approximation to the area of region \(A\). Give your answer correct to 3 decimal places. [4]
3. Deduce an approximation to the area of region \(B\). [2]

\includegraphics{figure_7} The diagram shows the curve with equation \(y = 2x - e^{\frac{1}{2}x}\). The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = 2\) and \(x = 4\).

1. Find the area of the shaded region, giving your answer in terms of e. [4]

The shaded region is rotated through four right angles about the \(x\)-axis.

1. Using Simpson's rule with two strips, estimate the volume of the solid formed. [5]

The function g is defined by $$g(x) = \mathrm{e}^{\sin x} \quad (0 \leq x \leq 2\pi)$$ The diagram below shows the graph of \(y = g(x)\) \includegraphics{figure_8}

1. Find the \(x\)-coordinate of each of the stationary points of the graph of \(y = g(x)\), giving your answers in exact form. [1 mark]
2. Use Simpson's rule with 3 ordinates to estimate $$\int_0^\pi g(x) \, \mathrm{d}x$$ giving your answer to two decimal places. [3 marks]
3. Explain how Simpson's rule could be used to find a more accurate estimate of the integral in part (b). [1 mark]

The diagram shows part of the graph of \(y = \cos^{-1} x\) \includegraphics{figure_7} The finite region enclosed by the graph of \(y = \cos^{-1} x\), the \(y\)-axis, the \(x\)-axis and the line \(x = 0.8\) is rotated by \(2\pi\) radians about the \(x\)-axis. Use Simpson's rule with five ordinates to estimate the volume of the solid formed. Give your answer to four decimal places. [5 marks]