The diagram shows the graph of $y={{x}^{2}}$ and $n$ rectangles, each with equal width, from \(x

Question

The diagram shows the graph of $y={{x}^{2}}$ and $n$ rectangles, each with equal width, from $x=0$ to $x=1$. Show that the total area of the $n$ rectangles can be written as $\frac{1}{{{n}^{3}}}\sum\limits_{r=1}^{n}{{{r}^{2}}}$.[2] Hence find ${\displaystyle \lim_{n\to\infty} \frac{1}{n^3} \sum_{r=1}^{n} r^2.}$[2]

Accepted Answer

(i) $\frac1{n^3}\sum_{r=1}^{n}r^2$ (shown) (ii) $\frac13$

2020 MI P2 Q1

2020 MI P2 Q1

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