The diagram below shows a sketch of the graph of $y = \sin^2 x$ for \(0 \le x \le \frac{\pi}{

Question

The diagram below shows a sketch of the graph of $y = \sin^2 x$ for $0 \le x \le \frac{\pi}{2}$. Rectangles each of width $\frac{\pi}{2n}$ are drawn under the curve for $0 \le x \le \frac{\pi}{2}$. Show that $A$, the total area of all the rectangles, is given by $a \sum_{k=1}^{n-1} \sin^2 \frac{k\pi}{2n}$, where $a$ is to be determined.[2] Find the exact value of $\lim_{n \to \infty} a \sum_{k=1}^{n-1} \sin^2 \frac{k\pi}{2n}$.[2] Hence, find the value of $\int_{0}^{1} \sin^{-1} \sqrt{y} \ \mathrm{d}y$.[2]

Accepted Answer

(a) $\frac{\pi }{4n}\sum\limits_{k=1}^{n-1}{\left( 1-\cos \left( \frac{k\pi }{n} \right) \right)}$ (b) $\frac{\pi }{4}$ (c) $\frac{\pi }{4}$

2024 YIJC P1 Q2

2024 YIJC P1 Q2