2
\includegraphics[max width=\textwidth, alt={}, center]{0f73e750-18a0-49ad-b4cb-fd6d14f0789e-2_531_700_395_719}
The diagram shows the curve \(y = \sqrt { } \left( 1 + 2 \tan ^ { 2 } x \right)\) for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\).
- Use the trapezium rule with three intervals to estimate the value of
$$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sqrt { } \left( 1 + 2 \tan ^ { 2 } x \right) \mathrm { d } x$$
giving your answer correct to 2 decimal places.
- The estimate found in part (i) is denoted by \(E\). Explain, without further calculation, whether another estimate found using the trapezium rule with six intervals would be greater than \(E\) or less than \(E\).