(i)     Express the area of the region $R$ as an integral with respect to $y$.

(ii)     Determine the area of $R$, giving your answer correct to four significant figures.

Find the exact volume generated when the region $R$ is rotated through $2\pi$ radians about the $y$-axis.

(i)     ${\text{area}} = \int_2^4 {\sqrt {y - 2} {\text{d}}y}$     M1A1

(ii)     $= 1.886{\text{ (4 sf only)}}$     A1

Note:     Award M0A0A1 for finding $1.886$ from $\int_0^{\sqrt 2 } {4 - f(x){\text{d}}x}$.

Award A1FT for a 4sf answer obtained from an integral involving $x$.

[3 marks]

${\text{volume}} = \pi \int_2^4 {(y - 2){\text{d}}y}$     (M1)

Note:     Award M1 for the correct integral with incorrect limits.

$= \pi \left[ {\frac{{{y^2}}}{2} - 2y} \right]_2^4$     (A1)

$= 2\pi {\text{ (exact only)}}$     A1

[3 marks]

Total [6 marks]