Use the substitution $x = \sin \theta$ to show that \(\displaystyle \displaystyle\int_{0}^{1}

Question

Use the substitution $x = \sin \theta$ to show that $\displaystyle \displaystyle\int_{0}^{1} \sqrt{1-x^{2}} \, \mathrm{d}x = \frac{\pi}{4}$.[4] The diagram shows the region $R$ bounded by the ellipse with equation $4x^{2} + (y-2)^{2} = 4$ and the line $y = 2x$. The ellipse and the line intersect at the origin and the point $(1, 2)$. Find the area of $R$, giving your answer in terms of $\pi$.[4] Find the volume of solid generated when $R$ is rotated $2\pi$ radians about the $x$-axis.[2]

Accepted Answer

(a) $\dfrac{\pi}{4}$ (b) $(\dfrac{\pi}{2}-1)\text{ units}^2$ (c) $(2\pi^2-\dfrac{16\pi}{3})\text{ units}^3\approx2.98\text{ units}^3$

2025 YIJC P1 Q11

2025 YIJC P1 Q11

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