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

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