<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Use the substitution \(x = \sin \theta\) to show that&nbsp;\(\int_{0}^{1} \sqrt{1-x^{2}} \, \mathrm{d}x = \frac{\pi}{4}\).</span><span class="inline-marks">[4]</span></div></li></ol>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1769316590327-ke92i38w72.png" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>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 \((1, 2)\).</p>
<ol class="abc-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Find the area of \(R\), giving your answer in terms of \(\pi\).</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the volume of solid generated when&nbsp;\(R\) is rotated \(2\pi\) radians about the \(x\)-axis.</span><span class="inline-marks">[2]</span></div></li></ol>

Use the substitution \(x = \sin \theta\) to show that \(\displaystyle \int_{0}^{1} \sqrt{1-x^{2}} \, \mathrm{d}x = \frac{\pi}{4}\).

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 \(\pi\).

Find the volume of solid generated when \(R\) is rotated \(2\pi\) radians about the \(x\)-axis.

2025 YIJC P1 Q12

2025 YIJC P1 Q12