OCR A Level: Further Pure 2 - Rectangle Approximation [January 2010 Q7]

Calculus Level pending

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

\((\text{i})\) 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 \((\text{iii})\).


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.
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