The graph of $y={{x}^{3}}+2$ for $0\le x\le 1$ is shown in the diagram. Rectangles, each of width $\

Question

The graph of $y={{x}^{3}}+2$ for $0\le x\le 1$ is shown in the diagram. Rectangles, each of width $\frac{1}{n}$, where $n$ is an integer, are drawn under the curve. Given that $\sum\limits_{r=1}^{n}{{{r}^{3}}}=\frac{{{n}^{2}}}{4}{{\left( n+1 \right)}^{2}}$, show that the total area of all $n$ rectangles, $A$, is given by $\frac{9}{4}-\frac{1}{2n}+\frac{1}{4{{n}^{2}}}$. [4] Deduce the exact area under the curve for $0\le x\le 1$. [1] Find the least value of $n$ such that the area $A$ is no less than $99\%$ of the exact area under the curve. [2]

Accepted Answer

$n = 22$

2011 DHS P1 Q11

2011 DHS P1 Q11

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