2024 YIJC P1 Q2
2024 YIJC P1 Q2
6 marks
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 \(\displaystyle \lim_{n\to\infty} a \sum_{k=1}^{n-1} \sin^2 \frac{k\pi}{2n}\).[2]
- Hence, find the value of \(\int_{0}^{1} \sin^{-1} \sqrt{y} \ \mathrm{d}y\).[2]
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Answer:(a) \(a=\dfrac{\pi}{2n}\), so \(A=\dfrac{\pi}{2n}\sum_{k=1}^{n-1}\sin^2\!\left(\dfrac{k\pi}{2n}\right)\) (b) \(\dfrac{\pi}{4}\) (c) \(\dfrac{\pi}{4}\)
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