Numerical integration comparison

5 The diagram shows a sketch of the graph of \(y = \sqrt { 27 + x ^ { 3 } }\). \includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-04_762_988_365_534}

1. The area of the shaded region, bounded by the curve, the \(x\)-axis and the lines \(x = 0\) and \(x = 4\), is given by \(\int _ { 0 } ^ { 4 } \sqrt { 27 + x ^ { 3 } } \mathrm {~d} x\). Use the mid-ordinate rule with five strips to find an estimate for this area. Give your answer to three significant figures.
2. With the aid of a diagram, explain whether the mid-ordinate rule applied in part (a) gives an estimate which is smaller than or greater than the area of the shaded region.
   (2 marks)

4. (a) Use the trapezium rule with two intervals of equal width to find an estimate for the value of the integral $$\int _ { 0 } ^ { 3 } e ^ { \cos x } d x$$ giving your answer to 3 significant figures.
(b) Use the trapezium rule with four intervals of equal width to find another estimate for the value of the integral to 3 significant figures.
(c) Given that the true value of the integral lies between the estimates made in parts (a) and (b), comment on the shape of the curve \(y = \mathrm { e } ^ { \cos x }\) in the interval \(0 \leq x \leq 3\) and explain your answer.
4. continued

6 The spreadsheet output in Fig. 6 shows approximations to \(\int _ { 0 } ^ { 1 } x ^ { - \sqrt { x } } \mathrm {~d} x\) found using the midpoint rule, denoted by \(M _ { n }\), and the trapezium rule, denoted by \(T _ { n }\). \begin{table}[h] \end{table}

1. Write down an efficient spreadsheet formula for cell C3.
2. By first completing the table in the Printed Answer Booklet using the Simpson's rule, calculate the most accurate estimate of \(\int _ { 0 } ^ { 1 } x ^ { - \sqrt { x } } \mathrm {~d} x\) that you can, justifying the precision quoted. \section*{END OF QUESTION PAPER}