Topic 5.07 · SL and HL

Trapezoidal rule: notes and practice questions

15 min read Jump to 18 practice questions

Summary

Trapezoidal Rule: numerical method to approximate area under a curve by dividing it into `n` trapezoids (strips).
`n` strips correspond to `n+1` function values ( $y_0, ..., y_n$ ).
Trapezoidal Rule formula: $\int_{a}^{b} f(x) dx \approx \frac{1}{2}h \left[ (y_0 + y_n) + 2(y_1 + y_2 + ... + y_{n-1}) \right]$
Width of each trapezoid: $h = \frac{b-a}{n}$
`a`: lower limit of integral; `b`: upper limit of integral; `n`: number of trapezoids.
`y_i`: function values calculated at each `x`-coordinate, where $y_0 = f(a)$ and $y_n = f(b)$ .
Percentage Error formula: $\text{Percentage Error} = \left| \frac{\text{Estimate} - \text{Exact}}{\text{Exact}} \right| \times 100$
Procedure:
Calculate $h = \frac{b-a}{n}$ .
Determine `x`-values starting from $x_0=a$ and adding `h` repeatedly.
Calculate corresponding `y`-values ( $y_i = f(x_i)$ ) for each `x`-value.
Substitute `h` and `y`-values into the trapezoidal rule formula.
If required, calculate percentage error, taking the absolute value of the result.
Show working (e.g., table of x/y values) for method marks.
Use GDC's numerical integration feature to find the exact area if not provided for percentage error calculations.
Avoid heavy rounding of `y`-values during intermediate steps to prevent compounding errors.
The principles, formulas, and methods are identical for IB Math AI SL and HL.

How it is examined

Distinctive to AI. Because it works on a table of data as well as on a function, it is the tool for an area with no formula, which is exactly the AI framing. The strip width $h$ has to be uniform, so a question with unequal intervals is out of syllabus. A frequent follow-up asks whether the estimate is an overestimate or an underestimate, which is answered from the concavity of the curve, and that is a reasoning mark rather than a calculation.

Given in the booklet

The trapezoidal rule, $A \approx \dfrac{1}{2}h\bigl((y_0 + y_n) + 2(y_1 + y_2 + \dots + y_{n-1})\bigr)$ , where $h = \dfrac{b - a}{n}$ .

Key ideas

Approximating areas using the trapezoidal rule.

Linking questions

Other contexts: irregular areas that are not described by mathematical functions, for example lakes.
Links to other subjects: kinematics (physics).
Use of technology: use dynamic graphing software to calculate the approximate area under a curve and interpret its meaning.
Enrichment only, so not examinable: other numerical integration techniques such as Simpson's rule.

Practice questions

18 questions · 1 easy · 15 medium · 2 hard

Showing 18 of 18

Question 1

EasyPaper 1 · calculator6 marks

(a)

Consider the function $f(x)$ with the following data points:

$a$ Estimate the area between the curve of $f(x)$ and the x-axis from $x = 0$ until x $= 6$ .

[2]

(b)

$b$ The equation of curve is $f(x) = \frac{1}{10}(x^{3} + 2x^{2} + 2x)$ , find the exact area between the curve of $f(x)$ and the x-axis from $x = 0$ until x $= 6$ .

[2]

(c)

$c$ Find the percentage error of the answer in part (a).

[2]

Question 2

MediumPaper 1 · calculator5 marks

(a)

A team of engineers is monitoring the flow of water into a large storage tank during a heavy rainfall. Let $F$ be the flow rate, in litres per hour, and $t$ be the time in hours since the monitoring began.

When $F$ is plotted against $t$ , the total volume of water collected in the tank is represented by the area between the graph and the horizontal $t$ -axis.

The flow rate, $F$ , is measured over the course of three hours. The results are shown in the following table.

$t$ (hours)	0	0.5	1.0	1.5	2.0	2.5	3.0
$F$ (litres/hour)	100	120	150	130	110	90	80

(a) Use the trapezoidal rule with an interval width of 0.5 hours to estimate the total volume of water collected in the tank during these three hours.

[3]

(b)

(b) The actual volume of water collected during these three hours was 340 litres.

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

[2]

Question 3

HardPaper 2 · calculator24 marks

(a)(i)

A landscape architect is designing a new public park. The northern boundary of the park is modelled by the function $g(x)$ , and other boundaries are straight lines. The plan view of the park is shown in the following diagram, where both axes represent distance and are measured in metres.

Graph showing a shaded region representing a park, bounded by a curve g(x) and straight lines, with points A, B, C, D.

The function $g(x)$ models the northern boundary of the park between points B and C and is given by

$g(x) = -\frac{1}{10}x^2 + 4x + 15$ , for $0 \le x \le 40$ .

(i) Find $g'(x)$ .

[2]

(a)(ii)

(ii) Hence find the coordinates of the point on the park's northern boundary that is furthest north.

[3]

(b)(i)

Point A has coordinates $(0, 0)$ , point B has coordinates $(0, 15)$ , point C has coordinates $(40, 15)$ and point D has coordinates $(40, 0)$ .

(i) Write down the integral which can be used to find the area of the shaded region representing the park.

[2]

(b)(ii)

(ii) Find the area of the park.

[2]

(c)(i)

(i) A landscaper uses the trapezoidal rule with 4 intervals to estimate the area of the park. Calculate this estimate.

[3]

(c)(ii)

(ii) Calculate the percentage error in the landscaper's estimate.

[3]

(c)(iii)

(iii) Suggest how the landscaper might be able to reduce the error whilst still using the trapezoidal rule.

[1]

(d)(i)

A square-shaped meditation garden, PQRS, is to be built within the park. The side [PS] lies on the southern boundary (x-axis), and the side [QR] lies on the eastern boundary of the park ( $x=40$ ). Point Q lies on the northern boundary curve $g(x)$ .

(i) Find the x-coordinate of point P for the largest area of the meditation garden.

[4]

(d)(ii)

(ii) Find the largest area of the meditation garden.

[4]

Question 4

MediumPaper 1 · calculator8 marks

(a)

The cross-section of a proposed tunnel entrance is modelled by a curve. The heights of the tunnel are measured at horizontal intervals and are given in the table below. All measurements are in cm.

Horizontal distance, x (cm)

Vertical distance, y (cm)

1.875

6.0

10.125

12.0

9.375

(a) Use the trapezoidal rule with $h = 10$ to find an approximation for the cross-sectional area of the tunnel entrance.

[2]

(b)(i)

(b)

It is given that the equation of the curve is $y = 0.0225x^2 - 0.000375x^3$ , for $0 \le x \le 60$ .

(i) Write down an integral to find the exact cross-sectional area.

[2]

(b)(ii)

(ii) Calculate the value of the cross-sectional area to two decimal places.

[2]

(c)

[2]

Question 5

HardPaper 2 · calculator12 marks

(a)

The diagram shows the cross-section of a newly constructed road tunnel. A coordinate system has been added with the origin, O, at the point where the tunnel meets the road surface on one side. All units are in metres.

Diagram showing the cross-section of a tunnel arch over a road surface, with a coordinate system. The arch starts at O(0,0) and ends at (10,0). The road level is at y=0. The arch goes above the road level.

To determine the air volume capacity, measurements of the tunnel's vertical height are taken every $2.5$ m from O. The heights are shown in the following table.

Horizontal distance from O in metres	0	2.5	5.0	7.5	10.0
Vertical height of tunnel in metres	0	3.2	4.8	3.5	0

Use the trapezoidal rule to find the cross-sectional area of the tunnel opening above the road surface.

[3]

(b)

A ventilation system pushes air through the tunnel at a rate of $0.8$ m s $^{-1}$ .

Find the volume of air that passes through the tunnel each second.

[2]

(c)

A maintenance vehicle, $3.0$ m tall, needs to perform inspections. Find the vertical clearance between the top of the vehicle and the tunnel ceiling at a horizontal distance of $2.5$ m from the entrance.

[1]

(d)

The curved arch of the tunnel can be modelled by the equation $y = -0.2x^2 + 2x$ , for $0 \le x \le 10$ .

Find the maximum height of the tunnel arch above the road surface.

[2]

(e)

A large cargo truck, $3.5$ m wide and $4.5$ m tall, needs to pass through the tunnel. The truck must stay in the centre of the road.

Determine whether the truck will be able to pass through the tunnel. Justify your answer.

[4]

Question 6

MediumPaper 1 · calculator5 marks

(a)

A city council is monitoring the rate at which water flows into a new reservoir during a storm. Let R be the rate, in thousands of litres per hour, at which water is flowing into the reservoir and t be the time in hours since the storm began.

When R is plotted against t, the total amount of water collected in the reservoir is represented by the area between the graph and the horizontal t-axis.

The rate, R, is measured over the course of three hours. The results are shown in the following table.

t (hours)	0	0.5	1.0	1.5	2.0	2.5	3.0
R (thousands of litres/hour)	10	25	40	35	20	15	30

The real amount of water collected during these three hours was 70 thousand litres.

Use the trapezoidal rule with an interval width of 0.5 hours to estimate the total amount of water collected during these three hours.

[3]

(b)

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

[2]

Question 7

MediumPaper 1 · calculator4 marks

A rocket engineer is designing a new nozzle. The internal shape of the nozzle is formed by rotating a curve about the central axis (y-axis).

The nozzle is 12 cm long. The internal radius of the nozzle is measured at 3 cm intervals along its length:

Length from base (cm)	Radius (cm)
0	5
3	4
6	3
9	2
12	1.5

Use the trapezoidal rule to estimate the internal volume of the nozzle.

Question 8

MediumPaper 1 · calculator8 marks

(a)

A civil engineer is surveying a river to estimate its cross-sectional area for flow rate calculations. The depth of the river is measured at regular horizontal intervals across its width, as shown in the table below.

Horizontal distance, x (m)	0	10	20	30	40	50	60
Depth, y (m)	0	2.5	8	13.5	16	12.5	0

Use the trapezoidal rule with $h = 10$ m to find an approximation for the cross-sectional area of the river.

[2]

(b)(i)

It is given that the equation of the curve modelling the riverbed cross-section is $y = 0.0005x^2(60-x)$ , for $0 \le x \le 60$ .

Write down an integral to find the exact cross-sectional area.

[2]

(b)(ii)

Calculate the value of the cross-sectional area to two decimal places.

[2]

(c)

Find the percentage error in the area found using the trapezoidal rule compared to the exact area.

[2]

Question 9

MediumPaper 1 · calculator10 marks

(a)

(a) A landscape architect is designing a curved retaining wall for a garden. The height of the wall, $h$ metres, is measured at $1$ m intervals along its horizontal length, $x$ metres, from one end. The measurements are given in the table below.

$x$ (m)	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$
$h$ (m)	$0$	$3.5$	$6$	$7.5$	$8$	$7.5$	$6$	$3.5$	$0$

Use the trapezium rule to estimate the cross-sectional area of the wall, giving your answer correct to $2$ decimal places.

[3]

(b)

(b) It is later discovered that the equation connecting $x$ and $h$ is $h(x) = 0.5x(8-x)$ .

Calculate the true value of the cross-sectional area of the wall.

[4]

(c)

[2]

(d)

(d) Explain why, in this case, the trapezium rule underestimates the true value.

[1]

Question 10

MediumPaper 1 · calculator4 marks

A research submarine is undergoing tests to measure its descent capabilities. During one test, its vertical velocity (rate of descent) is recorded at regular intervals, as shown in the table below.

Time ( $t$ in s)	Velocity ( $v$ in m/s)
0	0.0
10	2.5
20	6.0
30	11.5
40	18.0
50	25.5
60	34.0

Apply the trapezoidal rule to estimate the total vertical distance the submarine has descended by the time its velocity reaches $34.0$ m/s.

Question 11

MediumPaper 1 · calculator4 marks

(a) A scientist is studying the growth of a bacterial colony. The rate of growth, in arbitrary units, can be modelled by the function $f(x) = e^{0.5x}$ , where $x$ is the time in hours.

Estimate the total growth (area under the graph) of the colony from $x = 0$ to $x = 2$ hours using four trapezoids. Give your answer correct to four significant figures.

Question 12

MediumPaper 1 · calculator3 marks

A research drone is used to measure the speed of a river current at various distances from its bank. The data collected is presented in the table below.

Distance from bank ( $x$ m)	Speed of current ( $v(x)$ m/s)
0	0
2	3
4	5
6	4
8	0

(a) Use the trapezoidal rule to estimate the area under the graph of speed versus distance from $x=0$ m to $x=8$ m.

Question 13

MediumPaper 2 · calculator11 marks

(a)

(a) A chemical reaction produces a substance at a rate given by $R(t) = \frac{1}{t^2 + 1}$ grams per minute, where $t$ is the time in minutes. Estimate the total amount of substance produced (in grams) over the interval $0 \le t \le 2$ minutes using four trapezoids. Give your answer correct to four significant figures.

[4]

(b)(i)

(b) (i) Write down a definite integral that represents the exact total amount of substance produced.

[1]

(b)(ii)

(b) (ii) Hence, find the actual total amount of substance produced. Give your answer correct to four significant figures.

[3]

(c)

[3]

Question 14

MediumPaper 2 · calculator16 marks

(a)

A civil engineer is designing a cross-section for a new pedestrian tunnel. The shape of the tunnel's ceiling can be modelled by the curve $y = 10 - \frac{x^2}{4}$ , where $x$ and $y$ are measured in metres. The tunnel's base is along the $x$ -axis, and it starts from the $y$ -axis, extending to where the ceiling meets the $x$ -axis.

(a) Sketch the curve, showing the region representing the cross-sectional area of the tunnel. Shade this region.

[3]

(b)

(b) Using the trapezium rule with five strips, determine an approximation for the cross-sectional area of the tunnel.

[5]

(c)

[2]

(d)

(d) Using integration, determine the exact value of the cross-sectional area of the tunnel.

[4]

(e)

(e) Find the percentage error of your approximation from part (b) compared with the exact value found in part (d).

[2]

Question 15

MediumPaper 1 · calculator4 marks

The rate at which water flows into a reservoir, $R$ (in $m^3$ /hour), is measured at various times, $t$ (in hours), over an 8-hour period. The data is shown in the table below.

$t$ (hours)	$0$	$2$	$4$	$6$	$8$
$R$ ( $m^3$ /hour)	$10$	$15$	$12$	$18$	$8$

Estimate the total volume of water that flowed into the reservoir during this 8-hour period, using the trapezoidal rule.

Question 16

MediumPaper 1 · calculator18 marks

(a)

The diagram below shows a cross-section of a sculpted garden feature. The shaded area is bounded by the $x$ -axis, the vertical line $x = 6$ , the line $y = x-1$ and the curve $y = \frac{18}{x^2}$ .

Graph showing a shaded area bounded by x-axis, x=6, y=x-1 and y=18/x^2, with points P, Q, R, S labeled.

Using technology or otherwise, find the coordinates of points $P$ , $Q$ , $R$ and $S$ .

[4]

(b)

Approximate the area of the cross-section using the trapezoidal rule with 5 strips of equal width over the interval $[1,6]$ .

[6]

(c)

Explain why the approximation in part (b) is an overestimate.

[2]

(d)

Determine the exact area of the cross-section using integration.

[4]

(e)

Find the percentage error of your approximation from part (b), compared with the exact value from part (d).

[2]

Question 17

MediumPaper 2 · calculator13 marks

(a)(i)

(a) The depth, in metres, of a decorative planter box's cross-section is modelled by the function $y = -0.05x^3 + 0.6x^2$ , for $1 \le x \le 10$ , where $x$ is the horizontal distance in metres from the left edge of the box.

(i) Find $\frac{dy}{dx}$ .

[2]

(a)(ii)

(ii) Hence, find the maximum depth of the planter box.

[4]

(b)

(b) The depths of the planter box at $x=1$ , $x=4$ , $x=7$ , and $x=10$ are $0.55$ m, $6.4$ m, $12.25$ m, and $10.0$ m respectively.

Use the trapezoidal rule with three intervals to estimate the cross-sectional area of the planter box.

[3]

(c)(i)

(c)

(i) Write down the integral which can be used to find the exact cross-sectional area of the planter box.

[2]

(c)(ii)

(ii) Hence, find the exact cross-sectional area of the planter box.

[2]

Question 18

MediumPaper 2 · calculator12 marks

(a)

A landscape architect is designing a decorative arch for a garden entrance. The cross-section of the arch is modelled by the curve with equation $y = 0.5\sqrt{10x} - 0.5x + 1$ , for $0 \le x \le 10$ , where $x$ is the horizontal distance in metres from the start of the arch and $y$ is the height in metres above the ground. The architect wants to estimate the area under the arch.

Find the values of $a$ , $b$ and $c$ , shown in the table below, which represent the height of the arch at different horizontal distances.

x (m)	0	2.5	5	7.5	10
y (m)	1	a	b	c	1

[3]

(b)

The architect uses the trapezoidal rule with four intervals to estimate the area under the arch. Using the values from part (a), find this estimate of the area.

[3]

(c)(i)

Write down the integral that can be used to find the exact area under the arch.

[2]

(c)(ii)

Hence, use your graphic display calculator to find the exact area under the arch. Give your answer correct to one decimal place.

[2]

(d)

Calculate the percentage error of the architect's estimate in part (b) compared to the exact area found in part (c.ii).

[2]

Every Trapezoidal rule question, marked for you

Every answer is marked mark by mark, IB-style, and the AI tutor helps when you are stuck.

Where marks are lost

Leaving an answer in calculator notation. Never accepted in a final answer, and AI's constant calculator use makes this the easiest slip in the whole subject.

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What does Trapezoidal rule cover in IB Maths AI?

Trapezoidal Rule: numerical method to approximate area under a curve by dividing it into `n` trapezoids (strips). `n` strips correspond to `n+1` function values (y_0, ..., y_n). Trapezoidal Rule formula: ∫_a^b f(x) dx ≈ (1)/(2)h [ (y_0 + y_n) + 2(y_1 + y_2 + ... + y_n-1) ].

Is Trapezoidal rule SL or HL?

Both. SL and HL students study Trapezoidal rule to the same depth.

How do I revise Trapezoidal rule for IB Maths AI?

Start from the core idea: trapezoidal Rule: numerical method to approximate area under a curve by dividing it into `n` trapezoids (strips). In the exam: distinctive to AI. Because it works on a table of data as well as on a function, it is the tool for an area with no formula, which is exactly the AI framing. Then practise exam-style questions, easiest first, writing out every step of your working before you check it.

How does FourtyFive help me practise Trapezoidal rule?

FourtyFive has 18 Trapezoidal rule questions. Every answer you write is marked mark by mark, IB-style, and you see where each mark was won or lost. Every part has a hint, the AI tutor helps you through the step you are stuck on, and your Study Profile picks what to practise next.

Is FourtyFive free for Trapezoidal rule practice?

Yes. A free account gives you 50 marked answers a month, and you do not need a card to sign up.

Can I handwrite Trapezoidal rule answers on an iPad?

Yes. In the FourtyFive iPad app you write your working by hand with Apple Pencil, the way you would on paper, and it is marked the same way.

Trapezoidal rule: notes and practice questions

How it is examined

Linking questions

Practice questions

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16

Question 17

Question 18

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Where marks are lost

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How it is examined

Linking questions

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Question 3

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Question 9

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