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\includegraphics[max width=\textwidth, alt={}, center]{bca7281b-a6a9-4b4c-94e5-3da2a561ad86-08_663_1152_260_452} The diagram shows the curve with equation \(\mathrm { y } = 2 \mathrm { x } - \mathrm { x } ^ { 2 }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles of width \(\frac { 1 } { n }\).

1. By considering the sum of the areas of these rectangles, show that \(\int _ { 0 } ^ { 1 } \left( 2 x - x ^ { 2 } \right) d x < U _ { n }\), where $$U _ { n } = \left( 1 + \frac { 1 } { n } \right) \left( \frac { 2 } { 3 } - \frac { 1 } { 6 n } \right) .$$
2. Use a similar method to find, in terms of \(n\), a lower bound \(L _ { n }\) for \(\int _ { 0 } ^ { 1 } \left( 2 x - x ^ { 2 } \right) d x\).
3. Show that \(\lim _ { n \rightarrow \infty } \left( \mathrm { U } _ { n } - \mathrm { L } _ { \mathrm { n } } \right) = 0\).