22M.1.AHL.TZ2.14

pestleMathematicsAIHLPaper 122M· sl-5-8-trapezoid-rulesource ↗

The shape of a vase is formed by rotating a curve about the $y$ -axis.

The vase is $10 cm$ high. The internal radius of the vase is measured at $2 cm$ intervals along the height:

Use the trapezoidal rule to estimate the volume of water that the vase can hold.

Markscheme / solution

$V = π \int_{0}^{10} y^{2} d x$ OR $π \int_{0}^{10} x^{2} d y$ (M1)

$h = 2$

$\approx π \times \frac{1}{2} \times 2 \times ((4^{2} + 5^{2}) + 2 \times (6^{2} + 8^{2} + 7^{2} + 3^{2}))$ M1A1

$= 1120 {cm}^{3} (1121.548 \dots)$ A1

Note: Do not award the second M1 If the terms are not squared.

[4 marks]

Examiners’ report

This was a straightforward question on the trapezoidal rule, presented in an unfamiliar way, but only a tiny minority answered it correctly. It may be that candidates were introduced to the trapezium rule as an approximation to the area under a curve and here they were being asked to find an approximation to a volume and they were unable to see how that could be done.