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Date	May 2022	Marks available	2	Reference code	22M.1.SL.TZ1.6
Level	Standard Level	Paper	Paper 1	Time zone	Time zone 1
Command term	Find	Question number	6	Adapted from	N/A

Question

A modern art painting is contained in a square frame. The painting has a shaded region bounded by a smooth curve and a horizontal line.

When the painting is placed on a coordinate axes such that the bottom left corner of the painting has coordinates $(- 1, - 1)$ and the top right corner has coordinates $(2, 2)$ , the curve can be modelled by $y = f (x)$ and the horizontal line can be modelled by the $x$ -axis. Distances are measured in metres.

The artist used the equation $y = \frac{- x^{3} - 3 x^{2} + 4 x + 12}{10}$ to draw the curve.

Use the trapezoidal rule, with the values given in the following table, to approximate the area of the shaded region.

[3]

Find the exact area of the shaded region in the painting.

[2]

Find the area of the unshaded region in the painting.

[2]

Markscheme

$\frac{1}{2} (0.6 + 0 + 2 (1.2 + 1.2))$ (A1)(M1)

Note: Award A1 for evidence of $h = 1$ , M1 for a correct substitution into trapezoidal rule (allow for an incorrect $h$ only). The zero can be omitted in the working.

$2.7 m^{2}$ A1

[3 marks]

$\int_{- 1}^{2} \frac{- x^{3} - 3 x^{2} + 4 x + 12}{10} d x$ OR $\int_{- 1}^{2} f (x) d x$ (M1)

Note: Award M1 for using definite integration with correct limits.

$2.925 m^{2}$ A1

Note: Question requires exact answer, do not award final A1 for $2.93$ .

[2 marks]

$9 - 2.925$ (M1)

Note: Award M1 for $9$ seen as part of a subtraction.

$= 6.08 m^{2} (6.075)$ A1

[2 marks]

Examiners report

There seemed a better attempt at using the trapezium rule in this session compared to the two 2021 sessions. Despite many incorrect values for $h$ , candidates obtained the method mark for a correctly substituted formula (excluding $h$ ).

The exact answer of 2.925 was asked for in the question, yet candidates frequently rounded to three significant figures and hence lost the final mark.

Many candidates were able to correctly find the area of the unshaded region.

Syllabus sections

Topic 5—Calculus » SL 5.5—Integration introduction, areas between curve and x axis

Show 85 related questions

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22M.1.SL.TZ1.12b:
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Express this volume in ${\text{c}}{{\text{m}}^3}$ .
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Write down the coordinates of A.
19M.2.SL.TZ2.T_1b:
State the number of degrees of freedom.
19M.2.SL.TZ2.T_1c.i:
the expected frequency of female students who chose to take the Chinese class.
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State whether or not H₀ should be rejected. Justify your statement.
18M.2.SL.TZ1.S_4b:
On the grid above, sketch the graph of g for −2 ≤ x ≤ 4.
17M.2.SL.TZ2.S_8b.ii:
Write down the rate of change of $f$ at A.
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Find the probability that this adult is allergic to nuts and the liquid turns blue.
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Write down the probability that the second spin is yellow, given that the first spin is blue.
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Estimate the number of employees, from this 38, who are allergic to nuts.
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Find the coordinates of B.
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18M.2.SL.TZ2.S_3b:
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Find the minimum volume of empty space in the container during the 4 hour period.
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State in context what this value represents.
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Find $\int_{0}^{5} \frac{1000}{2 + t} d t$ .
18M.1.SL.TZ1.S_5b:
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17N.2.SL.TZ0.S_5b:
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Calculate the probability that one girl answered questions in Mandarin and the other answered questions in Hindi.
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16N.2.SL.TZ0.S_6a:
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21M.2.AHL.TZ1.2d.ii:
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Find the value of $p$ and of $q$ .
21M.1.SL.TZ2.13b:
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Describe how their profit changes if they increase production to over $30$ cars per month and up to $50$ cars per month. Justify your answer.
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Find the the rate of change of $f$ at B.
21M.2.SL.TZ1.5c.i:
Write down the integral which can be used to find the cross-sectional area of the tunnel.
21M.2.SL.TZ1.5c.ii:
Hence find the cross-sectional area of the tunnel.
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Write down the integral which can be used to find the cross-sectional area of the tunnel.
21N.1.SL.TZ0.13b:
Find the area of the shaded region in Irina’s design.
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Find the value of $t$ at which the ball hits the ground.
EXN.1.AHL.TZ0.10c:
Determine $\int_{0}^{365} P (t) d t$ and state what it represents.
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Find an expression for $P$ in terms of $t$ .
EXN.2.SL.TZ0.3a:
Find an expression for the height $h$ of the ball at time $t$ .
EXN.2.SL.TZ0.6g:
Use the expression found in (f) to calculate a value for $A$ .
21N.1.SL.TZ0.13a.i:
Write down the value of $c$ .

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