A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 11: Integration · Lesson 7

The trapezium rule

113 marks

Hints online, answers free with an account:
mathswariz.uk/worksheets/a-level-the-trapezium-rule/

Date

Do Now

Quick questions to start.

A1
Needed today
$f (x) = ln x$ , $x > 0$ . Find $f^{''} (x)$ and say whether the curve $y = ln x$ is concave or convex.
[2 marks]
Answer
A2
Common slip
Work out $0.6 sin 0.6$ , correct to $4$ decimal places.
[2 marks]
Answer
A3
Common slip
The interval $1 \leq x \leq 3$ is split into $4$ strips of equal width. Write down the width of each strip and the $x$ -values at the edges of the strips.
[2 marks]
Answer
A4
Needed today
A trapezium has parallel sides of length $2.4$ and $3.6$ , and the distance between them is $0.5$ . Find its area.
[2 marks]
Answer

Example, then your turn

A-Level

Your teacher works through each example with you. Then try the one beside it.

Example 1

Use the trapezium rule with $4$ strips to estimate $\int_{0}^{2} 1 + x^{2} d x$ , giving your answer to $3$ decimal places. The table shows values of $y = 1 + x^{2}$ .

x	0	0.5	1	1.5	2
y	1	1.118	1.4142	1.8028	2.2361

Hint

h = \frac{2 - 0}{4} = 0.5

. Work out the

5

ordinates, then use

\frac{h}{2} [y_{0} + y_{4} + 2 (y_{1} + y_{2} + y_{3})]

B1
Use the trapezium rule with $4$ strips to estimate $\int_{0}^{1} e^{x^{2}} d x$ , giving your answer to $3$ decimal places.
[4 marks]
Answer

Example 2

Use the trapezium rule with $4$ strips to estimate $\int_{1}^{2} \frac{1}{x} d x$ . Say whether your estimate is too big or too small, and find the percentage error.

Hint

f^{''} (x) = \frac{2}{x ^{3}} > 0

, so the curve is convex. Each chord lies above the curve. The exact value is

ln 2

B2
Use the trapezium rule with $3$ strips to estimate $\int_{1}^{4} x d x$ . Explain whether your estimate is too big or too small. Find the exact value and the percentage error.
[6 marks]
Answer

Example 3

Use the trapezium rule with $4$ strips to estimate $I = \int_{0}^{4} e^{x} d x$ . Hence estimate $\int_{0}^{4} (e^{x} + 3 x) d x$ and $\int_{0}^{4} e^{x + 1} d x$ .

Hint The extra

3 x

can be integrated exactly. And

e^{x + 1} = e \times e^{x}

B3
Use the trapezium rule with $4$ strips to estimate $I = \int_{0}^{2} ln (1 + x^{2}) d x$ . Hence estimate $\int_{0}^{2} ln ((1 + x^{2})^{3}) d x$ and $\int_{0}^{2} ln (e (1 + x^{2})) d x$ .
[5 marks]
Answer

Example 4

Use the trapezium rule with $4$ strips to estimate $\int_{0}^{π /2} x sin x d x$ . Find the exact value by integration by parts, and say how the estimate could be made more accurate.

Hint

h = \frac{π}{8}

. Keep the calculator in radians.

B4
Use the trapezium rule with $4$ strips to estimate $\int_{0}^{π /2} x cos x d x$ . Find the exact value and the percentage error.
[5 marks]
Answer

Practice

A-Level

C1
Mia uses the trapezium rule with $4$ strips to estimate $\int_{0}^{2} 2^{x} d x$ . She works out the ordinates at $x = 0, 0.5, 1, 1.5, 2$ correctly. Then she says: "There are $5$ ordinates, so $h = \frac{2}{5} = 0.4$ ," and gets $3.497$ . Explain her mistake. Find the correct estimate and say whether it is too big or too small.
[5 marks]
Answer

C2
Josh uses the trapezium rule with $2$ strips to estimate $\int_{0}^{2} e^{x} d x$ . He says: "The curve is increasing, so the estimate must be too small." Find his estimate and the exact value, and explain why he is wrong.
[5 marks]
Answer
C3
Ella uses the trapezium rule with $3$ strips to estimate $\int_{0}^{1.2} x sin x d x$ . Her table of values of $y = x sin x$ is shown. She gets $0.0106$ . Explain what has gone wrong and find the correct estimate.
[5 marks]
x 0 0.4 0.8 1.2
y 0 0.0028 0.0112 0.0251
Answer
C4
$\int_{0}^{1} \frac{4}{1 + x ^{2}} d x = π$ . Use the trapezium rule with $2$ strips and then with $4$ strips to estimate the integral. Find the percentage error in each, to $2$ decimal places, and comment on what happens when the number of strips is doubled.
[6 marks]
Answer

x	0	0.4	0.8	1.2
y	0	0.0028	0.0112	0.0251

Exam-style questions

Edexcel exam style

D1
The table shows values of $y = x ln x$ for $1 \leq x \leq 3$ , to $4$ decimal places.
x 1 1.5 2 2.5 3
y 0 ? 1.3863 ? 3.2958
(a)
Complete the table.
[2 marks]
Answer
(b)
Use the trapezium rule with all the values in the table to estimate $\int_{1}^{3} x ln x d x$ , giving your answer to $3$ decimal places.
[3 marks]
Answer
(c)
Use integration by parts to find the exact value of $\int_{1}^{3} x ln x d x$ , in the form $a ln 3 + b$ .
[4 marks]
Answer
(d)
Find the percentage error in your answer to part (b), and explain why the estimate is too big.
[2 marks]
Answer
Total for D1: 11 marks

x	1	1.5	2	2.5	3
y	0	?	1.3863	?	3.2958

D2
Amir uses the trapezium rule with $4$ strips to estimate $\int_{0}^{2} 4 + x^{2} d x$ . He writes: " $0.25 [2 (2) + 2 (2.8284) + 2.0616 + 2.2361 + 2.5] = 4.114$ . The curve bends upwards, so this is an underestimate."
Amir is wrong twice.
(a)
Explain the mistake in his use of the trapezium rule.
[2 marks]
(b)
Find the correct estimate, to $3$ decimal places.
[3 marks]
Answer
(c)
Given that $\frac{d ^{2}}{d x ^{2}} 4 + x^{2} = \frac{4}{( 4 + x ^{2} ) ^{3/2}}$ , explain whether the correct estimate is too big or too small.
[4 marks]
Total for D2: 9 marks
D3
The diagram shows the line $y = x + 1$ and the curve $y = x^{3} + 1$ . They meet at $x = 0$ and at $x = 2$ . Use the trapezium rule with $4$ strips to estimate the area of the shaded region between them, giving your answer to $2$ decimal places.
[7 marks]
Answer
D4
A cyclist starts from rest. Her speed $v$ m s $^{- 1}$ is recorded every $2$ seconds for $10$ seconds.
t (s) 0 2 4 6 8 10
v (m s⁻¹) 0.00 6.59 11.01 13.98 15.96 17.29
(a)
Use the trapezium rule with all the values in the table to estimate the distance she travels in the $10$ seconds.
[3 marks]
Answer
(b)
The speeds were taken from the model $v = 20 (1 - e^{- 0.2 t})$ . Show that the model gives a distance of $100 + 100 e^{- 2}$ m in the $10$ seconds.
[4 marks]
(c)
Explain why your answer to part (a) is smaller than the model's distance.
[2 marks]
Total for D4: 9 marks
D5
The curve $y = 1 - x^{2}$ for $0 \leq x \leq 1$ is a quarter of a circle of radius $1$ .
x 0 0.25 0.5 0.75 1
y 1 ? 0.866 ? 0
(a)
Complete the table, to $4$ decimal places.
[2 marks]
Answer
(b)
Use the trapezium rule with all the values in the table to estimate $\int_{0}^{1} 1 - x^{2} d x$ .
[3 marks]
Answer
(c)
Explain why the exact value of the integral is $\frac{π}{4}$ , and use your answer to part (b) to estimate $π$ .
[2 marks]
Answer
(d)
State whether this estimate of $π$ is too big or too small, giving a reason, and say how to make it more accurate.
[2 marks]
Total for D5: 9 marks
D6
$f (x) = \frac{3}{( x + 1 ) ( x + 4 )}$ , $x \geq 0$ .
(a)
Express $f (x)$ in partial fractions.
[3 marks]
Answer
(b)
Hence show that $\int_{0}^{2} f (x) d x = ln 2$ .
[3 marks]
(c)
Use the trapezium rule with $2$ strips to estimate the integral, and find the percentage error, to $2$ significant figures.
[3 marks]
Answer
Total for D6: 9 marks

t (s)	0	2	4	6	8	10
v (m s⁻¹)	0.00	6.59	11.01	13.98	15.96	17.29

x	0	0.25	0.5	0.75	1
y	1	?	0.866	?	0

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
Let $T_{n}$ be the trapezium rule estimate of $\int_{0}^{1} e^{x} d x$ using $n$ strips. Show that $T_{n} = \frac{e - 1}{2 n} \times \frac{e ^{1/ n} + 1}{e ^{1/ n} - 1}$ . Hence find the smallest $n$ for which $T_{n}$ is more than the exact value by less than $0.002$ .
[10 marks]
Hint 1 · What to try
The strips have width $h = \frac{1}{n}$ , and the ordinates are $e^{0}, e^{1/ n}, e^{2/ n}, \dots, e^{1}$ .
Hint 2 · The first line
Write $T_{n} = \frac{h}{2} (S + S^{'})$ , where $S = k = 0 \sum n - 1 e^{k / n}$ and $S^{'} = k = 1 \sum n e^{k / n} = e^{1/ n} S$ . $S$ is a geometric series.
Hint 3 · The full method
The exact value is $e - 1$ . Work out $T_{n} - (e - 1)$ for a few values of $n$ near where it drops below $0.002$ .

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can use the trapezium rule with n strips, taking h as the width of one strip.
I can say whether a trapezium rule estimate is too big or too small from whether the curve is convex or concave.
I can find the percentage error against an exact answer and say how to make an estimate better.
I can use one trapezium rule estimate to estimate a related integral.

Answers and mark scheme

The worked answers and the full mark scheme are free with a Mathswariz account. Teachers and students can both sign up.

Get the answers