2025 ASRJC P1 Q7

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 \(L\).

1. Use the substitution \(x = 2\sin\theta\) to show that the area of region \(A\) can be written in the expression \(\displaystyle \displaystyle\int_{b}^{a} 4\cos^2\theta \, \mathrm{d}\theta - c\), where \(a\), \(b\) and \(c\) are exact constants to be determined. Hence evaluate this area exactly.[6]
2. 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.[3]