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The diagram shows the curve with equation \(y = \sqrt [ 3 ] { x }\), together with a set of \(n\) rectangles of unit width.
- By considering the areas of these rectangles, explain why
$$\sqrt [ 3 ] { 1 } + \sqrt [ 3 ] { 2 } + \sqrt [ 3 ] { 3 } + \ldots + \sqrt [ 3 ] { n } > \int _ { 0 } ^ { n } \sqrt [ 3 ] { x } \mathrm {~d} x$$
- By drawing another set of rectangles and considering their areas, show that
$$\sqrt [ 3 ] { 1 } + \sqrt [ 3 ] { 2 } + \sqrt [ 3 ] { 3 } + \ldots + \sqrt [ 3 ] { n } < \int _ { 1 } ^ { n + 1 } \sqrt [ 3 ] { x } \mathrm {~d} x$$
- Hence find an approximation to \(\sum _ { n = 1 } ^ { 100 } \sqrt [ 3 ] { n }\), giving your answer correct to 2 significant figures.