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Date	May Specimen paper	Marks available	1	Reference code	SPM.1.SL.TZ0.10
Level	Standard Level	Paper	Paper 1	Time zone	Time zone 0
Command term	Find	Question number	10	Adapted from	N/A

Question

The following diagram shows part of the graph of $f(x) = \left( {6 - 3x} \right)\left( {4 + x} \right)$ , $x \in \mathbb{R}$ . The shaded region R is bounded by the $x$ -axis, $y$ -axis and the graph of $f$ .

Write down an integral for the area of region R.

[2]

Find the area of region R.

[1]

The three points A(0, 0) , B(3, 10) and C( $a$ , 0) define the vertices of a triangle.

Find the value of $a$ , the $x$ -coordinate of C, such that the area of the triangle is equal to the area of region R.

[2]

Markscheme

A = $\int_0^2 {\left( {6 - 3x} \right)\left( {4 + x} \right){\text{d}}x}$ A1A1

Note: Award A1 for the limits $x$ = 0, $x$ = 2. Award A1 for an integral of $f(x)$ .

[2 marks]

28 A1

[1 mark]

$28 = 0.5 \times a \times 10$ M1

$5.6\left( {\frac{{28}}{5}} \right)$ A1

[2 marks]

Examiners report

It was pleasing to see that, for those candidates who made a reasonable attempt at the paper, many were able to identify the correct values on the tree diagram.

[N/A]

Syllabus sections

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

Show 85 related questions

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Write down the value of a.
17N.2.SL.TZ0.T_4c:
Copy and complete the tree diagram.
19M.2.SL.TZ2.T_1a:
Write down the null hypothesis, H₀, for this test.
22M.1.SL.TZ2.6b.ii:
Calculate the area of this region.
18M.2.SL.TZ1.S_4a:
Write down the coordinates of the vertex of the graph of g.
SPM.1.SL.TZ0.10c:
The three points A(0, 0) , B(3, 10) and C( $a$ , 0) define the vertices of a triangle.

Find the value of $a$ , the $x$ -coordinate of C, such that the area of the triangle is equal to the area of region R.
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Find $\int {{{\left( {f\left( x \right)} \right)}^2}{\text{d}}x}$ .
17N.2.SL.TZ0.T_4f:
Find the probability that the tested adult is allergic to nuts given that the liquid turned blue.
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Write down the value of b.
21M.1.SL.TZ1.13b:
Sieun observes that when the angle is $40 °$ , the ball will travel a horizontal distance of $205.5 m$ .

Find an expression for the function $l (θ)$ .
EXN.2.SL.TZ0.3b.ii:
Hence write down the domain of $h$ .
22M.1.SL.TZ2.6b.i:
Write down an integral for the area of the shaded region in diagram 2.
22M.1.SL.TZ2.6c:
Hence, determine the area enclosed between $y = 6 - x$ and $y = 1.5 x^{2} - 2.5 x + 3$ .
22M.1.SL.TZ2.6a:
Calculate the area of the shaded region in diagram 1.
SPM.1.SL.TZ0.10a:
Write down an integral for the area of region R.
22M.1.SL.TZ1.12b:
One year after the company was founded, the profit was $4$ thousand dollars.

Find an expression for $P (t)$ , when $t \geq 0$ .
22M.1.SL.TZ1.6b:
Find the exact area of the shaded region in the painting.
22M.1.SL.TZ1.6c:
Find the area of the unshaded region in the painting.
22M.2.AHL.TZ1.2d:
The goat is tied in the field for $8$ hours.

Find the total volume of grass eaten by the goat during this time.
22M.2.AHL.TZ1.6d.i:
Find the area enclosed by $y = f (x)$ , the $x$ -axis and the line $x = 0.5$ .
21N.1.SL.TZ0.13a.iii:
Hence find the equation of the quadratic curve.
21N.1.SL.TZ0.13a.ii:
Hence form two equations in terms of $a$ and $b$ .
16N.2.SL.TZ0.T_6b:
Express this volume in ${\text{c}}{{\text{m}}^3}$ .
17M.2.SL.TZ2.S_8b.i:
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.
19M.2.SL.TZ2.T_1d:
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.
17N.2.SL.TZ0.T_4d:
Find the probability that this adult is allergic to nuts and the liquid turns blue.
19M.1.SL.TZ1.T_12c:
Write down the probability that the second spin is yellow, given that the first spin is blue.
17N.2.SL.TZ0.T_4g:
Estimate the number of employees, from this 38, who are allergic to nuts.
17M.2.SL.TZ2.S_8c.i:
Find the coordinates of B.
18M.3.AHL.TZ0.Hca_3c:
Hence write down a lower bound for $\sum\limits_{n = 4}^\infty {\frac{1}{{{n^3}}}}$ .
19M.2.SL.TZ2.T_1e.iii:
Find the probability that at least one of the two students is female.
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Find the probability that both spins are yellow.
16N.2.SL.TZ0.S_6b:
The empty barrel is being filled with water. The volume $V{\text{ }}{{\text{m}}^3}$ of water in the barrel after $t$ minutes is given by $V = 0.8(1 - {{\text{e}}^{ - 0.1t}})$ . How long will it take for the barrel to be half-full?
18M.2.SL.TZ2.S_3b:
The region enclosed by the graph of $f$ , the y-axis and the x-axis is rotated 360° about the x-axis.

Find the volume of the solid formed.
19M.2.SL.TZ2.S_2b:
The region enclosed by the graph of $f$ , the $x$ -axis and the $y$ -axis is rotated 360º about the $x$ -axis. Find the volume of the solid formed.
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Find the probability that at least one of the spins is yellow.
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Find the probability that the student does not take the Spanish class.
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Find the value of $p$ .
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Find the probability that both people chosen are not allergic to nuts.
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18N.2.SL.TZ0.S_10c:
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Find the minimum volume of empty space in the container during the 4 hour period.
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Find the probability that neither of the two students take the Spanish class.
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Find an upper bound for $\sum\limits_{n = 4}^\infty {\frac{1}{{{n^3}}}}$ .
17M.2.SL.TZ2.S_8d:
Let $R$ be the region enclosed by the graph of $f$ , the $x$ -axis, the line $x = b$ and the line $x = a$ . The region $R$ is rotated 360° about the $x$ -axis. Find the volume of the solid formed.
17M.2.SL.TZ2.S_8a:
Find the value of $p$ .
EXN.1.AHL.TZ0.10a.ii:
State in context what this value represents.
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Find the range of $h$ .
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Find the volume of the container.
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Find $\int_{0}^{5} \frac{1000}{2 + t} d t$ .
18M.1.SL.TZ1.S_5b:
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Find the probability that the liquid turns blue.
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Find the x-intercept of the graph of $f$ .
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During the interval $p$ < $t$ < $q$ , he volume of water in the container increases by $k$ m³. Find the value of $k$ .
17N.2.SL.TZ0.S_5b:
The following diagram shows part of the graph of $f$ .

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21M.1.SL.TZ2.13a:
Find an expression for $P$ in terms of $x$ .
18M.3.AHL.TZ0.Hca_3b:
Illustrate graphically the inequality $\sum\limits_{n = 5}^\infty {\frac{1}{{{n^3}}}} < \int\limits_4^\infty {\frac{1}{{{x^3}}}{\text{d}}x} < \sum\limits_{n = 4}^\infty {\frac{1}{{{n^3}}}}$ .
18M.1.SL.TZ2.T_7c:
Two girls are selected at random.

Calculate the probability that one girl answered questions in Mandarin and the other answered questions in Hindi.
21M.1.SL.TZ1.13a:
Determine whether the graph of $l$ against $θ$ is increasing or decreasing at $θ = 50 °$ .
21M.2.SL.TZ1.5a.i:
Find $\frac{d y}{d x}$ .
18M.1.SL.TZ2.S_2b:
Find the area of the region enclosed by the graph of $f$ , the x-axis and the lines x = 1 and x = 2 .
16N.2.SL.TZ0.S_6a:
Use the model to find the volume of the barrel.
21M.2.AHL.TZ1.2d.ii:
Hence find the cross-sectional area of the tunnel.
18N.2.SL.TZ0.S_10b.i:
Find the value of $p$ and of $q$ .
21M.1.SL.TZ2.13b:
The company regularly increases the number of cars it produces.

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.
17M.2.SL.TZ2.S_8c.ii:
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.
21M.2.AHL.TZ1.2d.i:
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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