<p> Curve \(C\) is a circle with radius \(2\) and center at the origin with equation \(x^2 + y^2 = 4\). <br>Line \(L\) has the equation \(\sqrt{3}y = x + 2\). The diagram below shows the shaded region \(A\) which is enclosed between \(C\) and \(L\).</p>
<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/1768620190985-op9batgzd1q.png" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Use the substitution \(x = 2\sin\theta\) to show that the area of region \(A\) can be written in the expression \(\int_{b}^{a} 4\cos^2\theta \, \mathrm{d}\theta - c\), where \(a\), \(b\) and \(c\) are exact constants to be determined. Hence evaluate this area exactly.</span><span class="inline-marks">[6]</span></div></li><li><div class="list-item-content"><span class="list-item-text">The region \(B\) is bounded by the curve \(C\), the line \(L\) and the line \(x = 2\). Find the volume generated when region \(B\) is rotated through \(2\pi\) radians about the \(y\)-axis. Give your answer to \(2\) decimal places.</span><span class="inline-marks">[3]</span></div></li></ol>

 Curve \(C\) is a circle with radius \(2\) and center at the origin with equation \(x^2 + y^2 = 4\). 
Line \(L\) has the equation \(\sqrt{3}y = x + 2\). The diagram below shows the shaded region \(A\) which is enclosed between \(C\) and \(L\).

Use the substitution \(x = 2\sin\theta\) to show that the area of region \(A\) can be written in the expression \(\displaystyle \int_{b}^{a} 4\cos^2\theta \, \mathrm{d}\theta - c\), where \(a\), \(b\) and \(c\) are exact constants to be determined. Hence evaluate this area exactly.

The region \(B\) is bounded by the curve \(C\), the line \(L\) and the line \(x = 2\). Find the volume generated when region \(B\) is rotated through \(2\pi\) radians about the \(y\)-axis. Give your answer to \(2\) decimal places.

2025 ASRJC P1 Q7

2025 ASRJC P1 Q7