Applications of integration: HCI 2024 Promo Q3

The diagram below shows a curve with equation $y=\sqrt{1-x}$. Let $A$ be the area bounded by the curve and the axes. A total of $n$ rectangles, each of equal width, are constructed to approximate the value of $A$.

(a)

Show that the total area of $n$ rectangles is given by $\displaystyle\frac1{n\sqrt n}\sum_{r=0}^{n-1}\sqrt{n-r}$. [2]

Suggested Solutions

(a)

Area of $n$ rectangles $$\begin{aligned} {}&= \frac{1}{n}\left(\sqrt{1 - \frac{0}{n}}\right) + \frac{1}{n}\left(\sqrt{1 - \frac{1}{n}}\right) + \dots + \frac{1}{n}\left(\sqrt{1 - \frac{n-1}{n}}\right)\\ &= \frac{1}{n}\left[\sqrt{\frac{n-0}{n}} + \sqrt{\frac{n-1}{n}} + \dots + \sqrt{\frac{n - (n-1)}{n}}\right]\\ &= \frac{1}{n}\sum_{r=0}^{n-1}\sqrt{\frac{n - r}{n}}\\ &= \frac{1}{n\sqrt{n}}\sum_{r=0}^{n-1}\sqrt{n - r}. \end{aligned}$$

(b)

Without using a calculator, evaluate $\displaystyle\lim_{n\to\infty}\frac1{n\sqrt n}\sum_{r=0}^{n-1}\sqrt{n-r}$. [3]

Suggested Solutions

(b)

$$\begin{aligned} \lim_{n\to\infty}\frac{1}{n\sqrt{n}}\sum_{r=0}^{n-1}\sqrt{n - r} &= \text{Area of infinitely many rectangles}\\ &= \text{Area under the curve from $x=0$ to $x=1$}\\ &= \int_0^1 \sqrt{1 - x}\,\mathrm{d} x\\ &= -\int_0^1 -\sqrt{1 - x}\,\mathrm{d} x\\ &= \left[-\frac{2}{3}\sqrt{(1 - x)^3}\right]_0^1\\ &= \frac{2}{3}. \end{aligned}$$