2026 EJC P2 Q2

The curve \(C\) has equation \(y=\dfrac{\cos x}{\sqrt{1+\sin x}}\) for \(0\leq x\leq\dfrac\pi2\). The region \(R\) is bounded by the curve \(C\), the \(x\)-axis, and the lines \(x=0\) and \(x=\dfrac\pi6\).

1. Use the substitution \(u=1+\sin x\) to find the exact area of \(R\).[4]

The region \(S\) is bounded by the curve \(C\), the line \(y=\dfrac1{\sqrt2}\), and the \(y\)-axis.

2. Verify that \(y=\dfrac1{\sqrt2}\) when \(x=\dfrac\pi6\). Hence find the volume of the solid formed when \(S\) is rotated through \(2\pi\) radians about the \(x\)-axis.[3]