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 \(\frac { \pi } { 4 }\).
| \(x\) | 0 | \(\frac { \pi } { 16 }\) | \(\frac { \pi } { 8 }\) | \(\frac { 3 \pi } { 16 }\) | \(\frac { \pi } { 4 }\) |
| \(y\) | 1 | | | 1.20269 | |
(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).