Find the coordinates of the points where the line meets the curve.

The region \(S\) is rotated by \(2\pi \) about the \(x\)-axis to generate a solid.

(i)     Write down an integral that represents the volume \(V\) of the solid.

(ii)     Find the volume \(V\).

(a)     \(\frac{\pi }{2}(1.57),{\text{ }}\frac{{3\pi }}{2}(4.71)\)     A1A1

hence the coordinates are \(\left( {\frac{\pi }{2},{\text{ }}\frac{\pi }{2}} \right),{\text{ }}\left( {\frac{{3\pi }}{2},{\text{ }}\frac{{3\pi }}{2}} \right)\)     A1

[3 marks]

(i)     \(\pi \int_{\frac{\pi }{2}}^{\frac{{3\pi }}{2}} {\left( {{x^2} - {{(x + 2\cos x)}^2}} \right){\text{d}}x} \)     A1A1A1

Note:     Award A1 for \({x^2} - {(x + 2\cos x)^2}\), A1 for correct limits and A1 for \(\pi \).

(ii)     \(6{\pi ^2}{\text{ }}( = 59.2)\)     A2

Notes:     Do not award ft from (b)(i).

[5 marks]