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

**Calculus**Level pending

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