Trapezium rule then exact integration comparison

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. 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.

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

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

The graph of \(y = \frac{2x^3}{x^2 + 1}\) is shown for \(0 \leq x \leq 4\) Caroline is attempting to approximate the shaded area, A, under the curve using the trapezium rule by splitting the area into \(n\) trapezia.

1. When \(n = 4\)
   1. State the number of ordinates that Caroline uses. [1 mark]
   2. Calculate the area that Caroline should obtain using this method. Give your answer correct to two decimal places. [3 marks]
2. Show that the exact area of \(A\) is $$16 - \ln 17$$ Fully justify your answer. [5 marks]
3. Explain what would happen to Caroline's answer to part (a)(ii) as \(n \to \infty\) [1 mark]

The following algorithm produces a numerical approximation for the integral $$I = \int_A^B x^4 \, dx$$

Step 1	Start
Step 2	Input the values of A, B and N
Step 3	Let H = (B - A) / N
Step 4	Let C = H / 2
Step 5	Let D = 0
Step 6	Let D = D + A\(^4\) + B\(^4\)
Step 7	Let E = A
Step 8	Let E = E + H
Step 9	If E = B go to Step 12
Step 10	Let D = D + 2 × E\(^4\)
Step 11	Go to Step 8
Step 12	Let F = C × D
Step 13	Output F
Step 14	Stop

For the case when A = 1, B = 3 and N = 4,

1. 1. complete the table in the answer book to show the results obtained at each step of the algorithm.
   2. State the final output. [4]
2. Calculate, to 3 significant figures, the percentage error between the exact value of \(I\) and the value obtained from using the approximation to \(I\) in this case. [3]