The diagram below shows a sketch of the curve $y=\frac{x}{\sqrt{1+{{x}^{2}}}}$ for $x\ge 0$

Question

The diagram below shows a sketch of the curve $y=\frac{x}{\sqrt{1+{{x}^{2}}}}$ for $x\ge 0$. Rectangles, each of width $\frac{1}{n}$, where $n\in {{\mathbb{Z}}^{+}}$, are drawn under the curve for $0\le x\le 1$. Show that the total area of all the rectangles, $A$, can be written as $\frac{1}{n}\sum\limits_{r=1}^{n-1}{\frac{r}{\sqrt{{{n}^{2}}+{{r}^{2}}}}}$.[3] Find the exact value of $\mathop{\lim}\limits_{n\to \infty }\,A$.[3]

Accepted Answer

(a) $A=\dfrac1n\sum_{r=1}^{n-1}\dfrac{r}{\sqrt{n^2+r^2}}$ (b) $\sqrt2-1$

2023 CJC P1 Q2

2023 CJC P1 Q2

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