19M.2.AHL.TZ1.H_7

pestleMathematicsAIHLPaper 219M· ahl-5-12-areas-under-a-curve-onto-x-or-y-axis-volumes-of-revolution-about-x-and-ysource ↗

The function $f$ is defined by $f\left( x \right) = {\left( {x - 1} \right)^2}$ , $x$ ≥ 1 and the function $g$ is defined by $g\left( x \right) = {x^2} + 1$ , $x$ ≥ 0.

The region $R$ is bounded by the curves $y = f\left( x \right)$ , $y = g\left( x \right)$ and the lines $y = 0$ , $x = 0$ and $y = 9$ as shown on the following diagram.

The shape of a clay vase can be modelled by rotating the region $R$ through 360˚ about the $y$ -axis.

Find the volume of clay used to make the vase.

Markscheme / solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

volume $= \pi {\int_0^9 {\left( {{y^{\frac{1}{2}}} + 1} \right)} ^2}{\text{d}}y - \pi \int_1^9 {\left( {y - 1} \right)} {\text{d}}y$ (M1)(M1)(M1)(A1)(A1)

Note: Award (M1) for use of formula for rotating about $y$ -axis, (M1) for finding at least one inverse, (M1) for subtracting volumes, (A1)(A1)for each correct expression, including limits.

$= 268.6 \ldots - 100.5 \ldots \left( {85.5\pi - 32\pi } \right)$

$= 168\left( { = 53.5\pi } \right)$ A2

[7 marks]

Examiners’ report

[N/A]