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\includegraphics[max width=\textwidth, alt={}, center]{d421652f-576d-4843-abbf-54404e225fec-10_1015_988_260_577} The diagram shows the curve with equation \(\mathrm { y } = ( 1 - \mathrm { x } ) ^ { 2 }\) 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 \(\int _ { 0 } ^ { 1 } ( 1 - x ) ^ { 2 } d x < U _ { n }\), where $$U _ { n } = \frac { 2 n ^ { 2 } + 3 n + 1 } { 6 n ^ { 2 } }$$
2. Use a similar method to find, in terms of \(n\), a lower bound \(L _ { n }\) for \(\int _ { 0 } ^ { 1 } ( 1 - x ) ^ { 2 } d x\).
3. Show that \(\lim _ { n \rightarrow \infty } \left( U _ { n } - L _ { n } \right) = 0\).