5
\includegraphics[max width=\textwidth, alt={}, center]{34177829-f05d-449e-8881-5ab4d852c4ce-3_458_643_285_751}
The diagram shows the part of the curve \(y = x \mathrm { e } ^ { - x }\) for \(0 \leqslant x \leqslant 2\), and its maximum point \(M\).
- Find the \(x\)-coordinate of \(M\).
- Use the trapezium rule with two intervals to estimate the value of
$$\int _ { 0 } ^ { 2 } x \mathrm { e } ^ { - x } \mathrm {~d} x$$
giving your answer correct to 2 decimal places.
- State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the integral in part (ii).