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\includegraphics[max width=\textwidth, alt={}, center]{c342b622-a560-46da-9e64-edc4b7b3be93-2_662_1063_986_484} The diagram shows the curve \(y = \mathrm { e } ^ { - \frac { 1 } { x } }\) for \(0 < x \leqslant 1\). A set of ( \(n - 1\) ) rectangles is drawn under the curve as shown.

1. Explain why a lower bound for \(\int _ { 0 } ^ { 1 } \mathrm { e } ^ { - \frac { 1 } { x } } \mathrm {~d} x\) can be expressed as $$\frac { 1 } { n } \left( \mathrm { e } ^ { - n } + \mathrm { e } ^ { - \frac { n } { 2 } } + \mathrm { e } ^ { - \frac { n } { 3 } } + \ldots + \mathrm { e } ^ { - \frac { n } { n - 1 } } \right)$$
2. Using a set of \(n\) rectangles, write down a similar expression for an upper bound for \(\int _ { 0 } ^ { 1 } \mathrm { e } ^ { - \frac { 1 } { x } } \mathrm {~d} x\).
3. Evaluate these bounds in the case \(n = 4\), giving your answers correct to 3 significant figures.
4. When \(n \geqslant N\), the difference between the upper and lower bounds is less than 0.001 . By expressing this difference in terms of \(n\), find the least possible value of \(N\).