2023 JPJC P1 Q10

On the same axes, sketch the graphs of \(y=\sin 2x\) and \(y=\cos x\) for \(0\le x\le \frac{\pi }{2}\), stating the exact coordinates of the points where the curves cross the axes.[2]

Solve exactly the inequality \(\sin 2x>\cos x\), where \(0\le x\le \frac{\pi }{2}\).[2]

Hence, find \(\displaystyle\displaystyle \displaystyle\int_{0}^{\frac{\pi }{2}}{\left| \sin 2x-\cos x \right|}\ \mathrm{d}x\) without using a calculator.[4]

The region bounded by the curves \(y=\sin 2x\), \(y=\cos x\) and the \(y\)-axis, where \(x\ge 0\), is rotated through \(2\pi \) radians about the \(x\)-axis. Find the exact volume of the solid obtained[3]