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Date	May 2022	Marks available	4	Reference code	22M.1.AHL.TZ2.14
Level	Additional Higher Level	Paper	Paper 1	Time zone	Time zone 2
Command term	Estimate	Question number	14	Adapted from	N/A

Question

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

$V = π \int_{0}^{10} y^{2} d x$ 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.

Syllabus sections

Topic 5—Calculus » SL 5.8—Trapezoid rule

Show 21 related questions

EXM.3.AHL.TZ0.7a.ii:
With reference to the shape of the graph, explain whether your answer to part (a)(i) will be an over-estimate or an underestimate of the area.
EXM.2.AHL.TZ0.12c.ii:
Find the value of this area.
EXM.3.AHL.TZ0.7b.ii:
Write down the coefficient of determination.
22M.1.SL.TZ1.6a:
Use the trapezoidal rule, with the values given in the following table, to approximate the area of the shaded region.
EXM.2.AHL.TZ0.12b.ii:
Write down the coefficient of determination.
EXM.3.AHL.TZ0.7c.ii:
Find the value of this area.
EXM.3.AHL.TZ0.7d.ii:
Hence explain how a straight line graph could be drawn using the coordinates in the table.
EXM.2.AHL.TZ0.12b.i:
Use all the coordinates in the table to find the equation of the least squares cubic regression curve.
EXM.2.AHL.TZ0.12a:
Use the trapezoidal rule to find an estimate for the area.
EXM.3.AHL.TZ0.7a.i:
Use the trapezoidal rule to find an estimate for the area.
EXM.3.AHL.TZ0.7b.i:
Use all the coordinates in the table to find the equation of the least squares cubic regression curve.
EXM.2.AHL.TZ0.12c.i:
Write down an expression for the area enclosed by the cubic regression curve, the $x$ -axis, the $y$ -axis and the line $x = 10$ .
EXM.3.AHL.TZ0.7d.i:
Show that ${\text{ln}}\,y = qx + {\text{ln}}\,p$ .
EXM.3.AHL.TZ0.7d.iii:
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EXM.3.AHL.TZ0.7d.iv:
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21M.2.AHL.TZ1.2c:
Use the trapezoidal rule, with three intervals, to estimate the cross-sectional area of the tunnel.
21M.2.SL.TZ1.5b:
Use the trapezoidal rule, with three intervals, to estimate the cross-sectional area of the tunnel.
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Write down an expression for the area enclosed by the cubic function, the $x$ -axis, the $y$ -axis and the line $x = 4.4$ .
EXN.2.SL.TZ0.6b:
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21N.1.SL.TZ0.6a:
Use the trapezoidal rule with an interval width of $0.4$ to estimate the total amount of carbon dioxide emitted during these two hours.
21N.1.SL.TZ0.6b:
The real amount of carbon dioxide emitted during these two hours was $72$ tonnes.

Find the percentage error of the estimate found in part (a).

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Topic 5—Calculus » AHL 5.12—Areas under a curve onto x or y axis. Volumes of revolution about x and y

Topic 5—Calculus

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