Error analysis for approximation

6. \begin{figure}[h] \end{figure} Figure 1 shows the graph of the curve with equation $$y = \frac { 16 } { x ^ { 2 } } - \frac { x } { 2 } + 1 , \quad x > 0$$ The finite region \(R\), bounded by the lines \(x = 1\), the \(x\)-axis and the curve, is shown shaded in Figure 1. The curve crosses the \(x\)-axis at the point \(( 4,0 )\).

1. Complete the table with the values of \(y\) corresponding to \(x = 2\) and 2.5

   \(x\)	1	1.5	2	2.5	3	3.5	4
   \(y\)	16.5	7.361			1.278	0.556	0

2. Use the trapezium rule with all the values in the completed table to find an approximate value for the area of \(R\), giving your answer to 2 decimal places.
3. Use integration to find the exact value for the area of \(R\).

9. \(y\) \begin{figure}[h] \end{figure} The finite region \(R\), as shown in Figure 2, is bounded by the \(x\)-axis and the curve with equation $$y = 27 - 2 x - 9 \sqrt { } x - \frac { 16 } { x ^ { 2 } } , \quad x > 0$$ The curve crosses the \(x\)-axis at the points \(( 1,0 )\) and \(( 4,0 )\).

1. Complete the table below, by giving your values of \(y\) to 3 decimal places.

   \(x\)	1	1.5	2	2.5	3	3.5	4
   \(y\)	0	5.866		5.210		1.856	0

2. Use the trapezium rule with all the values in the completed table to find an approximate value for the area of \(R\), giving your answer to 2 decimal places.
3. Use integration to find the exact value for the area of \(R\).

12. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve \(C\) with equation $$y = \frac { x ^ { 2 } \ln x } { 3 } - 2 x + 4 , \quad x > 0$$ The finite region \(S\), shown shaded in Figure 4, is bounded by the curve \(C\), the \(x\)-axis and the lines with equations \(x = 1\) and \(x = 3\)

1. Complete the table below with the value of \(y\) corresponding to \(x = 2\). Give your answer to 4 decimal places.

   \(x\)	1	1.5	2	2.5	3
   \(y\)	2	1.3041		0.9089	1.2958

2. Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an estimate for the area of \(S\), giving your answer to 3 decimal places.
3. Use calculus to find the exact area of \(S\). Give your answer in the form \(\frac { a } { b } + \ln c\), where \(a , b\) and \(c\) are integers.
4. Hence calculate the percentage error in using your answer to part (b) to estimate the area of \(S\). Give your answer to one decimal place.
5. Explain how the trapezium rule could be used to obtain a more accurate estimate for the area of \(S\).

7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation $$y = \frac { x + 7 } { \sqrt { 2 x - 3 } } \quad x > \frac { 3 } { 2 }$$ The region \(R\), shown shaded in Figure 1, is bounded by the curve, the line with equation \(x = 4\), the \(x\)-axis and the line with equation \(x = 6\)

1. Use the trapezium rule with 4 strips of equal width to find an estimate for the area of \(R\), giving your answer to 2 decimal places.
2. Using the substitution \(u = 2 x - 3\), or otherwise, use calculus to find the exact area of \(R\), giving your answer in the form \(a + b \sqrt { 5 }\), where \(a\) and \(b\) are constants to be found.

13. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve with equation \(y = 12 x ^ { 2 } \ln \left( 2 x ^ { 2 } \right) , x > 0\) The finite region \(R\), shown shaded in Figure 4, is bounded by the curve, the line with equation \(x = 1\), the \(x\)-axis and the line with equation \(x = 2\) The table below shows corresponding values of \(x\) and \(y\) for \(y = 12 x ^ { 2 } \ln \left( 2 x ^ { 2 } \right)\), with the values of \(y\) given to 3 significant figures.

1. Use the trapezium rule, with all the values of \(y\), to obtain an estimate for the area of \(R\), giving your answer to 2 significant figures.
2. Use the substitution \(u = x ^ { 2 }\) to show that the area of \(R\) is given by $$\int _ { 1 } ^ { 4 } 6 u ^ { \frac { 1 } { 2 } } \ln ( 2 u ) \mathrm { d } u$$
3. Hence, using calculus, find the exact area of \(R\), writing your answer in the form \(a + b \ln 2\), where \(a\) and \(b\) are constants to be found.

5. \begin{figure}[h] \end{figure} Figure 1 shows the graph of the curve with equation $$y = x \mathrm { e } ^ { 2 x } , \quad x \geqslant 0$$ The finite region \(R\) bounded by the lines \(x = 1\), the \(x\)-axis and the curve is shown shaded in Figure 1.

1. Use integration to find the exact value for the area of \(R\).
2. Complete the table with the values of \(y\) corresponding to \(x = 0.4\) and 0.8 .

   \(x\)	0	0.2	0.4	0.6	0.8	1
   \(y = x \mathrm { e } ^ { 2 x }\)	0	0.29836		1.99207		7.38906

3. Use the trapezium rule with all the values in the table to find an approximate value for this area, giving your answer to 4 significant figures.

2. (a) Given that \(y = \sec x\), complete the table with the values of \(y\) corresponding to \(x = \frac { \pi } { 16 } , \frac { \pi } { 8 }\) and \(\frac { \pi } { 4 }\). (b) Use the trapezium rule, with all the values for \(y\) in the completed table, to obtain an estimate for \(\int _ { 0 } ^ { \frac { \pi } { 4 } } \sec x \mathrm {~d} x\). Show all the steps of your working, and give your answer to 4 decimal places. The exact value of \(\int _ { 0 } ^ { \frac { \pi } { 4 } } \sec x \mathrm {~d} x\) is \(\ln ( 1 + \sqrt { } 2 )\).
(c) Calculate the \% error in using the estimate you obtained in part (b).

8. $$I = \int _ { 0 } ^ { 5 } \mathrm { e } ^ { \sqrt { } ( 3 x + 1 ) } \mathrm { d } x$$

1. Given that \(y = \mathrm { e } ^ { \sqrt { } ( 3 x + 1 ) }\), complete the table with the values of \(y\) corresponding to \(x = 2\), 3 and 4.

   \(x\)	0	1	2	3	4	5
   \(y\)	\(\mathrm { e } ^ { 1 }\)	\(\mathrm { e } ^ { 2 }\)				\(\mathrm { e } ^ { 4 }\)

2. Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an estimate for the original integral \(I\), giving your answer to 4 significant figures.
3. Use the substitution \(t = \sqrt { } ( 3 x + 1 )\) to show that \(I\) may be expressed as \(\int _ { a } ^ { b } k t e ^ { t } \mathrm {~d} t\), giving the values of \(a , b\) and \(k\).
4. Use integration by parts to evaluate this integral, and hence find the value of \(I\) correct to 4 significant figures, showing all the steps in your working.

2. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = x \ln x , x \geqslant 1\). The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis and the line \(x = 4\). The table shows corresponding values of \(x\) and \(y\) for \(y = x \ln x\).

1. Complete the table with the values of \(y\) corresponding to \(x = 2\) and \(x = 2.5\), giving your answers to 3 decimal places.
2. Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an estimate for the area of \(R\), giving your answer to 2 decimal places.
3. 1. Use integration by parts to find \(\int x \ln x \mathrm {~d} x\).
   2. Hence find the exact area of \(R\), giving your answer in the form \(\frac { 1 } { 4 } ( a \ln 2 + b )\), where \(a\) and \(b\) are integers.

7. $$I = \int _ { 2 } ^ { 5 } \frac { 1 } { 4 + \sqrt { } ( x - 1 ) } \mathrm { d } x$$

1. Given that \(y = \frac { 1 } { 4 + \sqrt { } ( x - 1 ) }\), complete the table below with values of \(y\) corresponding to \(x = 3\) and \(x = 5\). Give your values to 4 decimal places.

   \(x\)	2	3	4	5
   \(y\)	0.2		0.1745

2. Use the trapezium rule, with all of the values of \(y\) in the completed table, to obtain an estimate of \(I\), giving your answer to 3 decimal places.
3. Using the substitution \(x = ( u - 4 ) ^ { 2 } + 1\), or otherwise, and integrating, find the exact value of \(I\).

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \frac { x } { 1 + \sqrt { } x }\). The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis, the line with equation \(x = 1\) and the line with equation \(x = 4\).

1. Complete the table with the value of \(y\) corresponding to \(x = 3\), giving your answer to 4 decimal places.
   (1)

   \(x\)	1	2	3	4
   \(y\)	0.5	0.8284		1.3333

2. Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an estimate of the area of the region \(R\), giving your answer to 3 decimal places.
3. Use the substitution \(u = 1 + \sqrt { } x\), to find, by integrating, the exact area of \(R\).
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a98d4a7f-1e6d-4294-9b5c-c945e8fbe83e-07_743_1568_219_182} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} Figure 2 shows a sketch of part of the curve \(C\) with parametric equations $$x = 1 - \frac { 1 } { 2 } t , \quad y = 2 ^ { t } - 1$$ The curve crosses the \(y\)-axis at the point \(A\) and crosses the \(x\)-axis at the point \(B\).

7. \begin{figure}[h] \end{figure} Figure 1 shows part of the curve with equation \(y = \sqrt { } ( \tan x )\). The finite region \(R\), which is bounded by the curve, the \(x\)-axis and the line \(x = \frac { \pi } { 4 }\), is shown shaded in Figure 1.

1. Given that \(y = \sqrt { } ( \tan x )\), complete the table with the values of \(y\) corresponding to \(x = \frac { \pi } { 16 } , \frac { \pi } { 8 }\) and \(\frac { 3 \pi } { 16 }\), giving your answers to 5 decimal places.

   \(x\)	0	\(\frac { \pi } { 16 }\)	\(\frac { \pi } { 8 }\)	\(\frac { 3 \pi } { 16 }\)	\(\frac { \pi } { 4 }\)
   \(y\)	0				1

2. Use the trapezium rule with all the values of \(y\) in the completed table to obtain an estimate for the area of the shaded region \(R\), giving your answer to 4 decimal places. The region \(R\) is rotated through \(2 \pi\) radians around the \(x\)-axis to generate a solid of revolution.
3. Use integration to find an exact value for the volume of the solid generated. \section*{LO}

7. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve with equation \(y = x ^ { \frac { 1 } { 2 } } \ln 2 x\).
The finite region \(R\), shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 4\)

1. Use the trapezium rule, with 3 strips of equal width, to find an estimate for the area of \(R\), giving your answer to 2 decimal places.
2. Find \(\int x ^ { \frac { 1 } { 2 } } \ln 2 x \mathrm {~d} x\).
3. Hence find the exact area of \(R\), giving your answer in the form \(a \ln 2 + b\), where \(a\) and \(b\) are exact constants.

5. \begin{figure}[h] \end{figure} Figure 1 shows part of the curve with equation \(x = 4 t \mathrm { e } ^ { - \frac { 1 } { 3 } t } + 3\). The finite region \(R\) shown shaded in Figure 1 is bounded by the curve, the \(x\)-axis, the \(t\)-axis and the line \(t = 8\).

1. Complete the table with the value of \(x\) corresponding to \(t = 6\), giving your answer to 3 decimal places.

   \(t\)	0	2	4	6	8
   \(x\)	3	7.107	7.218		5.223

2. Use the trapezium rule with all the values of \(x\) in the completed table to obtain an estimate for the area of the region \(R\), giving your answer to 2 decimal places.
3. Use calculus to find the exact value for the area of \(R\).
4. Find the difference between the values obtained in part (b) and part (c), giving your answer to 2 decimal places.

2. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = x \mathrm { e } ^ { - \frac { 1 } { 2 } x } , x \geqslant 0\).
The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis, and the line \(x = 4\). The table shows corresponding values of \(x\) and \(y\) for \(y = x e ^ { - \frac { 1 } { 2 } x }\).

1. Complete the table with the value of \(y\) corresponding to \(x = 2\)
2. Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an estimate for the area of \(R\), giving your answer to 2 decimal places.
3. 1. Find \(\int x \mathrm { e } ^ { - \frac { 1 } { 2 } x } \mathrm {~d} x\).
   2. Hence find the exact area of \(R\), giving your answer in the form \(a + b \mathrm { e } ^ { - 2 }\), where \(a\) and \(b\) are integers.

2. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = x ^ { 2 } \ln x , x \geqslant 1\) The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis and the line \(x = 2\) The table below shows corresponding values of \(x\) and \(y\) for \(y = x ^ { 2 } \ln x\)

1. Complete the table above, giving the missing value of \(y\) to 4 decimal places.
2. Use the trapezium rule with all the values of \(y\) in the completed table to obtain an estimate for the area of \(R\), giving your answer to 3 decimal places.
3. Use integration to find the exact value for the area of \(R\).

8. The finite region \(R\) is bounded by the curve \(y = 1 + 3 \sqrt { x }\), the \(x\)-axis and the lines \(x = 2\) and \(x = 8\).

1. Use the trapezium rule with three intervals, each of width 2 , to estimate to 3 significant figures the area of \(R\).
2. Use integration to find the exact area of \(R\) in the form \(a + b \sqrt { 2 }\).
3. Find the percentage error in the estimate made in part (a).

7. \begin{figure}[h] \end{figure} Diagram not drawn to scale Figure 1 shows a sketch of part of the curve with equation \(y = \frac { 1 } { \sqrt { 2 x + 5 } } , x > - 2.5\) The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis and the lines with equations \(x = 2\) and \(x = 5\)

1. Use the trapezium rule with three strips of equal width to find an estimate for the area of \(R\), giving your answer to 3 decimal places.
2. Use calculus to find the exact area of \(R\).
3. Hence calculate the magnitude of the error of the estimate found in part (a), giving your answer to one significant figure.

11. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve with equation $$y = ( \ln x ) ^ { 2 } \quad x > 0$$ The finite region \(R\), shown shaded in Figure 2, is bounded by the curve, the line with equation \(x = 2\), the \(x\)-axis and the line with equation \(x = 4\) The table below shows corresponding values of \(x\) and \(y\), with the values of \(y\) given to 4 decimal places.

1. Use the trapezium rule, with all the values of \(y\) in the table, to obtain an estimate for the area of \(R\), giving your answer to 3 significant figures.
2. Use algebraic integration to find the exact area of \(R\), giving your answer in the form $$y = a ( \ln 2 ) ^ { 2 } + b \ln 2 + c$$ where \(a\), \(b\) and \(c\) are integers to be found.

9. The finite region \(R\) is bounded by the curve \(y = 1 + 3 \sqrt { x }\), the \(x\)-axis and the lines \(x = 2\) and \(x = 8\).

1. Use the trapezium rule with three intervals of equal width to estimate to 3 significant figures the area of \(R\).
2. Use integration to find the exact area of \(R\) in the form \(a + b \sqrt { 2 }\).
3. Find the percentage error in the estimate made in part (a).

3. A student tests the accuracy of the trapezium rule by evaluating \(I\), where $$I = \int _ { 0.5 } ^ { 1.5 } \left( \frac { 3 } { x } + x ^ { 4 } \right) \mathrm { d } x$$

1. Complete the student's table, giving values to 2 decimal places where appropriate.

   \(x\)	0.5	0.75	1	1.25	1.5
   \(\frac { 3 } { x } + x ^ { 4 }\)	6.06	4.32

2. Use the trapezium rule, with all the values from your table, to calculate an estimate for the value of \(I\).
3. Use integration to calculate the exact value of \(I\).
4. Verify that the answer obtained by the trapezium rule is within \(3 \%\) of the exact value.

14 The graph of \(y = \frac { 2 x ^ { 3 } } { x ^ { 2 } + 1 }\) is shown for \(0 \leq x \leq 4\) \includegraphics[max width=\textwidth, alt={}, center]{6b1312f4-9a5c-4465-8129-7d37e99efefe-20_1022_640_411_701} Caroline is attempting to approximate the shaded area, \(A\), under the curve using the trapezium rule by splitting the area into \(n\) trapezia. 14

1. When \(n = 4\) 14
2. 1. State the number of ordinates that Caroline uses. 14
3. (ii) Calculate the area that Caroline should obtain using this method.
   Give your answer correct to two decimal places.
   14
4. Show that the exact area of \(A\) is $$16 - \ln 17$$ Fully justify your answer.
   

   14

   Explain what would happen to Caroline's answer to part (a)(ii) as \(n \rightarrow \infty\)

   [1 mark]

   Do not write outside the box

   \includegraphics[max width=\textwidth, alt={}]{6b1312f4-9a5c-4465-8129-7d37e99efefe-23_2488_1716_219_153}