The diagram shows a sketch of the curve $y=\mathrm{f}(x)$. The shaded region under the curve betwe

Question

The diagram shows a sketch of the curve $y=\mathrm{f}(x)$. The shaded region under the curve between $x=0$ and $x=1$, as shown in the diagram, is $A$. This region is split into $n$ vertical strips of equal width, $h$.

State the value of $h$.[1]
Hence show, using a sketch, that $\displaystyle \sum_{r=1}^{n}h\mathrm{f}(rh)$ is less than the area of $A$.[2]

You are now given that $\displaystyle \mathrm{f}(x)=\frac{e^{-2x}}{1+e^{-2x}}$.

Hence find the exact value of $\displaystyle \lim_{n\to\infty}\sum_{r=1}^{n}h\mathrm{f}(rh)$.
[3]

Accepted Answer

(a) $h=\frac1n$ (b) right-endpoint sum is less than $A$ (c) $\frac12\ln\left(\frac{2e^2}{e^2+1}\right)$

2026 ACJC P1 Q6

2026 ACJC P1 Q6

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