2022 HCI Promo Q9
(i)
The diagram shows a sketch of the curve ${{C}_{1}}$ with equation $\text{f}\left( x \right)=x{{\text{e}}^{{{x}^{2}}}}$ for $0\le x\le 1$.
The region under ${{C}_{1}}$ is given by $A$.
(a) By sketching suitable vertical rectangular strips of equal width $\frac{1}{n}$ on the diagram, determine whether
${{S}_{n}}=\frac{1}{n}\left[ \text{f}\left( \frac{1}{n} \right)+\text{f}\left( \frac{2}{n} \right)+\text{f}\left( \frac{3}{n} \right)+…+\text{f}\left( \frac{n}{n} \right) \right]$
is more than or less than the area of $A$.
[2]
(b) Find the exact value of $\underset{n\to \infty }{\mathop{\lim }}\,{{S}_{n}}$.
[2]
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