Show trapezium rule gives specific value

4

1. Use the trapezium rule with three intervals to show that the value of \(\int _ { 1 } ^ { 4 } \ln x \mathrm {~d} x\) is approximately \(\ln 12\).
2. Use a graph of \(y = \ln x\) to show that \(\ln 12\) is an under-estimate of the true value of \(\int _ { 1 } ^ { 4 } \ln x \mathrm {~d} x\).

6

1. Use the trapezium rule, with 2 strips each of width 4 , to show that an approximate value of \(\int _ { 1 } ^ { 9 } 4 \sqrt { x } \mathrm {~d} x\) is \(32 + 16 \sqrt { 5 }\).
2. Use a sketch graph to explain why the actual value of \(\int _ { 1 } ^ { 9 } 4 \sqrt { x } \mathrm {~d} x\) is greater than \(32 + 16 \sqrt { 5 }\).
3. Use integration to find the exact value of \(\int _ { 1 } ^ { 9 } 4 \sqrt { x } \mathrm {~d} x\).

1. A continuous curve has equation \(y = \mathrm { f } ( x )\).

The table shows corresponding values of \(x\) and \(y\) for this curve, where \(a\) and \(b\) are constants. The trapezium rule is used, with all the \(y\) values in the table, to find an approximate area under the curve between \(x = 3\) and \(x = 4\) Given that this area is 17.59

1. show that \(a + 2 b = 51\) Given also that the sum of all the \(y\) values in the table is 97.2
2. find the value of \(a\) and the value of \(b\)

6 Fig. 6 shows a partially completed spreadsheet.
This spreadsheet uses the trapezium rule with four strips to estimate \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sqrt { 1 + \sin x } \mathrm {~d} x\). \begin{table}[h] \end{table}

1. Show how the value in cell B3 is calculated.
2. Show how the values in cells D2 to D6 are used to calculate the value in cell E7.
3. Complete the calculation to estimate \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sqrt { 1 + \sin x } \mathrm {~d} x\). Give your answer to 3 significant figures.

\includegraphics{figure_4} The diagram shows the part of the curve \(y = 3x \sin 2x\) for which \(0 \leqslant x \leqslant \frac{1}{2}\pi\). The maximum point on the curve is denoted by \(P\).

1. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(\tan 2x + 2x = 0\). [3]
2. Use the Newton-Raphson method, with a suitable initial value, to find the \(x\)-coordinate of \(P\), giving your answer correct to 4 decimal places. Show the result of each iteration. [4]
3. The trapezium rule, with four strips of equal width, is used to find an approximation to $$\int_0^{\frac{1}{2}\pi} 3x \sin 2x \, dx.$$ Show that the result can be expressed as \(k\pi^2(\sqrt{2} + 1)\), where \(k\) is a rational number to be determined. [4]
4. 1. Evaluate \(\int_0^{\frac{1}{2}\pi} 3x \sin 2x \, dx\). [1]
   2. Hence determine whether using the trapezium rule with four strips of equal width gives an under- or over-estimate for the area of the region enclosed by the curve \(y = 3x \sin 2x\) and the \(x\)-axis for \(0 \leqslant x \leqslant \frac{1}{2}\pi\). [1]
   3. Explain briefly why it is not easy to tell from the diagram alone whether the trapezium rule with four strips of equal width gives an under- or over-estimate for the area of the region in this case. [1]