Find k .

The shaded region is rotated $360^\circ$ about the x-axis. Let V be the volume of the solid formed.

Write down an expression for V .

The shaded region is rotated $360^\circ$ about the x-axis. Let V be the volume of the solid formed.

Find V .

evidence of valid approach     (M1)

e.g. $y = 0$ , $\sin x = 0$

$2\pi  = 6.283185 \ldots$

$k = 6.28$     A1     N2

[2 marks]

attempt to substitute either limits or the function into formula     (M1)

(accept absence of ${\rm{d}}x$ )

e.g. $V = \pi \int_\pi ^k {{{(f(x))}^2}{\rm{d}}x}$ , $\pi \int {{{((x - 1)\sin x)}^2}}$ , $\pi \int_\pi ^{6.28 \ldots } {{y^2}{\rm{d}}x}$

correct expression     A2     N3

e.g. $\pi \int_\pi ^{6.28} {{{(x - 1)}^2}{{\sin }^2}x{\rm{d}}x}$ , $\pi \int_\pi ^{2\pi } {{{((x - 1)\sin x)}^2}{\rm{d}}x}$ 

[3 marks]

$V = 69.60192562 \ldots$

$V = 69.6$     A2     N2

[2 marks]

Candidates showed marked improvement in writing fully correct expressions for a volume of revolution. Common errors of course included the omission of dx , using the given domain as the upper and lower bounds of integration, forgetting to square their function and/or the omission of $\pi$ . There were still many who were unable to use their calculator successfully to find the required volume.

Candidates showed marked improvement in writing fully correct expressions for a volume of revolution. Common errors of course included the omission of dx , using the given domain as the upper and lower bounds of integration, forgetting to square their function and/or the omission of $\pi$ . There were still many who were unable to use their calculator successfully to find the required volume.

Candidates showed marked improvement in writing fully correct expressions for a volume of revolution. Common errors of course included the omission of  dx, using the given domain as the upper and lower bounds of integration, forgetting to square their function and/or the omission of $\pi$ . There were still many who were unable to use their calculator successfully to find the required volume.