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.