The diagram shows the curve \(y=\sqrt [ 3 ]{ x } \), together with a set of \(n\) rectangles of unit width.

###### There are **2** marks available for part (i), **3** marks for part (ii) and **3** marks for part (iii).

###### In total, this question is worth **11.1%** of all available marks in the paper.

###### This is part of the set OCR A Level Problems.

\((\text{i})\) By considering the areas of these rectangles, explain why
\[\sqrt [ 3 ]{ 1 }+\sqrt [ 3 ]{ 2 }+\sqrt [ 3 ]{ 3 }+\cdots+\sqrt [ 3 ]{ n } > \int _{ 0 }^{ n }{ \sqrt [ 3 ]{ x } } \, dx .\]
\((\text{ii})\) By drawing another set of rectangles and considering their areas, show that
\[\sqrt [ 3 ]{ 1 }+\sqrt [ 3 ]{ 2 }+\sqrt [ 3 ]{ 3 }+\cdots+\sqrt [ 3 ]{ n } < \int _{ 1 }^{ n+1 }{ \sqrt [ 3 ]{ x } } \, dx .\]
\((\text{iii})\) Hence find an approximation to \(\displaystyle \sum _{ n=1 }^{ 100 }{ \sqrt [ 3 ]{ n } } \), giving your answer correct to *2 significant figures*.

**Input your answer to part \((\text{iii})\).**

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