sketch the graph of $f$;

show that ${\left( {f(x)} \right)^2} = \frac{3}{2} + 2\sin x - \frac{1}{2}\cos 2x$;

find the volume of the solid formed when the graph of f is rotated through $2\pi$ radians about the x-axis.

    A1

[1 mark]

${(1 + \sin x)^2} = 1 + 2\sin x + {\sin ^2}x$

$= 1 + 2\sin x + \frac{1}{2}(1 - \cos 2x)$     A1

$= \frac{3}{2} + 2\sin x - \frac{1}{2}\cos 2x$     AG

[1 mark]

$V = \pi \int_0^{\frac{{3\pi }}{2}} {{{(1 + \sin x)}^2}{\text{d}}x}$     (M1)

$= \pi \int_0^{\frac{{3\pi }}{2}} {\left( {\frac{3}{2} + 2\sin x - \frac{1}{2}\cos 2x} \right){\text{d}}x}$

$= \pi \left[ {\frac{3}{2}x - 2\cos x - \frac{{\sin 2x}}{4}} \right]_0^{\frac{{3\pi }}{2}}$     A1

$= \frac{{9{\pi ^2}}}{4} + 2\pi$     A1A1

[4 marks]

Parts (a) and (b) were almost invariably correctly answered by candidates. In (c), most errors involved the integration of $\cos (2x)$ and the insertion of the limits.

Parts (a) and (b) were almost invariably correctly answered by candidates. In (c), most errors involved the integration of $\cos (2x)$ and the insertion of the limits.

Parts (a) and (b) were almost invariably correctly answered by candidates. In (c), most errors involved the integration of $\cos (2x)$ and the insertion of the limits.