A-level · Numerical methods

The trapezium ruleExam-style questions

Estimating a definite integral by slicing the area into trapezium-topped strips: the formula (h/2)[y₀ + yₙ + 2(y₁ + … + y_(n−1))], choosing h, and deciding whether the estimate is an over- or under-estimate from the curve's shape.

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28 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
The curve $C$ has equation

$y = 4 - 2^{x}$
(a) Sketch $C$ . Mark on your sketch the coordinates of the two points where $C$ crosses the coordinate axes, and label the asymptote to $C$ with its equation.

(3 marks)

The finite region $R$ is bounded by $C$ , the $x$ -axis and the $y$ -axis.

The table below gives values of $y$ , to 4 decimal places where appropriate, at points on $C$ .

$\begin{array}{c|cccccc} x & 0 & 0.4 & 0.8 & 1.2 & 1.6 & 2 \\ \hline y & 3 & 2.6805 & 2.2589 & 1.7026 & 0.9686 & 0 \end{array}$
(b) Use the trapezium rule with all the values of $y$ in the table to find an estimate for the area of $R$ . Give your answer to 2 decimal places.

(3 marks)
(c) Using your answer to part (b) and making your method clear, estimate

(i) $\displaystyle\int_{0}^{2} \left(2^{x} + x^{3}\right)\mathrm{d}x$

(ii) $\displaystyle\int_{0}^{2} \left(2^{x+1} - 8\right)\mathrm{d}x$

(3 marks)
Level 4 · 9 marksTry it
2
Figure 1

Figure 1 shows the cross-section of a solid concrete ramp in a skate park. The cross-section is bounded by the horizontal ground, a vertical back wall and the curved riding surface of the ramp.

At a horizontal distance of $x$ metres from the back wall, the riding surface is $y$ metres above the ground. The table below shows values of $y$ measured at intervals of $0.4$ m.

$\begin{array}{c|ccccccc} x & 0 & 0.4 & 0.8 & 1.2 & 1.6 & 2.0 & 2.4 \\ \hline y & 1.20 & 0.76 & 0.46 & 0.25 & 0.11 & 0.03 & 0 \end{array}$
(a) Use the trapezium rule with all the values of $y$ in the table to estimate the area of the cross-section of the ramp.

(3 marks)

The ramp is $2.5$ m wide and has the same cross-section throughout its width. The concrete has density $2400\ \text{kg m}^{-3}$ .
(b) Hence estimate the mass of the ramp. Give your answer in kilograms to 2 significant figures.

(2 marks)
(c) Is your answer to part (b) likely to be an overestimate or an underestimate of the mass of the ramp? Give a reason for your answer.

(1 mark)
Level 4 · 6 marksTry it
3
The table gives values of $y = f(x)$ .

$\begin{array}{c|ccccc} x & 0 & 0.6 & 1.2 & 1.8 & 2.4 \\\\ \hline y & 3.2 & 4.1 & 4.6 & 4.0 & 2.5 \end{array}$

Use the trapezium rule with all five values to estimate $\displaystyle\int_0^{2.4} y\,dx$ .
Level 4 · 3 marksTry it
4
(a) Use the trapezium rule with 4 strips to estimate $\displaystyle\int_2^6 \ln x\,dx$ to 3 decimal places.
(b) State, with a reason, whether this under- or over-estimates the integral.
Level 4 · 5 marksTry it
5
$I = \int_1^3 \sqrt{1 + x^3}\;\mathrm{d}x$

Figure 1 shows the region whose area is given by $I$ .Figure 1
(a) Use the trapezium rule with four strips of equal width to find an estimate for $I$ , giving your answer to $4$ decimal places. (4 marks)
(b) State, with a reason, whether your answer to part (a) is an overestimate or an underestimate of $I$ . (2 marks)
Level 4 · 6 marksTry it

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