Trapezium rule accuracy improvement explanation

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

1. Use the trapezium rule with all the values of \(y\) in the table, to obtain an approximation for the area of \(R\), giving your answer to 2 decimal places.
2. Explain how the trapezium rule can be used to give a more accurate approximation for the area of \(R\).
3. Use calculus, showing each step in your working, to obtain an exact value for the area of \(R\). Give your answer in its simplest form.

2. \includegraphics[max width=\textwidth, alt={}, center]{27703044-8bb3-4809-9454-ae6774fec060-1_485_808_973_520} The diagram shows the curve with equation \(y = \sqrt { 4 x - 1 }\).

1. Use the trapezium rule with four intervals of equal width to estimate the area of the shaded region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 3\).
2. Explain briefly how you could use the trapezium rule to obtain a more accurate estimate of the area of the shaded region.

2

1. Use the trapezium rule, with 3 strips each of width 3 , to estimate the area of the region bounded by the curve \(y = \sqrt [ 3 ] { 7 + x }\), the \(x\)-axis, and the lines \(x = 1\) and \(x = 10\). Give your answer correct to 3 significant figures.
2. Explain how the trapezium rule could be used to obtain a more accurate estimate of the area.

2

1. Use the trapezium rule, with 4 strips each of width 1.5, to estimate the value of $$\int _ { 4 } ^ { 10 } \sqrt { 2 x - 1 } \mathrm {~d} x ,$$ giving your answer correct to 3 significant figures.
2. Explain how the trapezium rule could be used to obtain a more accurate estimate.

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.

14. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve \(C\) with equation $$y = \frac { x ^ { 2 } \ln x } { 3 } - 2 x + 5 , \quad x > 0$$ The finite region \(S\), shown shaded in Figure 4, is bounded by the curve \(C\), the line with equation \(x = 1\), the \(x\)-axis and the line with equation \(x = 3\) The table below shows corresponding values of \(x\) and \(y\) with the values of \(y\) given to 4 decimal places as appropriate.

1. Use the trapezium rule, with all the values of \(y\) in the table, to obtain an estimate for the area of \(S\), giving your answer to 3 decimal places.
2. Explain how the trapezium rule could be used to obtain a more accurate estimate for the area of \(S\).
3. Show that the exact area of \(S\) can be written in the form \(\frac { a } { b } + \ln c\), where \(a , b\) and \(c\) are integers to be found.
   (In part c, solutions based entirely on graphical or numerical methods are not acceptable.)

1. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \frac { x } { 1 + \sqrt { x } } , x \geqslant 0\) The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the line with equation \(x = 1\), the \(x\)-axis and the line with equation \(x = 3\) The table below shows corresponding values of \(x\) and \(y\) for \(y = \frac { x } { 1 + \sqrt { } x }\)

1. Use the trapezium rule, with all the values of \(y\) in the table, to find an estimate for the area of \(R\), giving your answer to 3 decimal places.
2. Explain how the trapezium rule can be used to give a better approximation for the area of \(R\).
3. Giving your answer to 3 decimal places in each case, use your answer to part (a) to deduce an estimate for
   1. \(\int _ { 1 } ^ { 3 } \frac { 5 x } { 1 + \sqrt { x } } \mathrm {~d} x\)
   2. \(\int _ { 1 } ^ { 3 } \left( 6 + \frac { x } { 1 + \sqrt { x } } \right) \mathrm { d } x\)