| Date | November 2015 | Marks available | 3 | Reference code | 15N.2.hl.TZ0.5 |
| Level | HL only | Paper | 2 | Time zone | TZ0 |
| Command term | Determine and Express | Question number | 5 | Adapted from | N/A |
Question
A function is defined by \(f(x) = {x^2} + 2,{\text{ }}x \ge 0\). A region \(R\) is enclosed by \(y = f(x)\),the \(y\)-axis and the line \(y = 4\).
(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.
Markscheme
(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]
