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Date	May Example question	Marks available	2	Reference code	EXM.2.AHL.TZ0.12
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Find	Question number	12	Adapted from	N/A

Question

The curve $y = f\left( x \right)$ is shown in the graph, for $0 \leqslant x \leqslant 10$ .

The curve $y = f\left( x \right)$ passes through the following points.

It is required to find the area bounded by the curve, the $x$ -axis, the $y$ -axis and the line $x = 10$ .

One possible model for the curve $y = f\left( x \right)$ is a cubic function.

Use the trapezoidal rule to find an estimate for the area.

[3]

Use all the coordinates in the table to find the equation of the least squares cubic regression curve.

[3]

b.i.

Write down the coefficient of determination.

[1]

b.ii.

Write down an expression for the area enclosed by the cubic regression curve, the $x$ -axis, the $y$ -axis and the line $x = 10$ .

[1]

c.i.

Find the value of this area.

[2]

c.ii.

Markscheme

Area = $\frac{2}{2}\left( {2 + 2\left( {4.5 + 4.2 + 3.3 + 4.5} \right) + 8} \right)$ M1A1

Area = 43 A1

[3 marks]

$y = 0.0389{x^3} - 0.534{x^2} + 2.06x + 2.06$ M1A2

[3 marks]

b.i.

${R^2} = 0.991$ A1

[1 mark]

b.ii.

Area = $\int\limits_0^{10} {y\,dx}$ A1

[1 mark]

c.i.

42.5 A2

[2 marks]

c.ii.

Examiners report

[N/A]

b.i.

[N/A]

b.ii.

[N/A]

c.i.

[N/A]

c.ii.

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.3.AHL.TZ0.7b.ii:
Write down the coefficient of determination.
22M.1.AHL.TZ2.14:
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.
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:
By finding the equation of a suitable regression line, show that $p = 1.83$ and $q = 0.986$ .
EXM.3.AHL.TZ0.7d.iv:
Hence find the area enclosed by the exponential function, the $x$ -axis, the $y$ -axis and the line $x = 4.4$ .
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.
EXM.3.AHL.TZ0.7c.i:
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:
By dividing the area between the curve and the $x$ ‐axis into two trapezoids of unequal width show that $A > 14.5$ , justifying the direction of the inequality.
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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