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]