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
A curve has equation $y = \sqrt[3]{x^2 + 5}$

The table below gives the value of $y$ , to 3 decimal places, at six equally spaced values of $x$ .

$\begin{array}{c|cccccc} x & 1 & 1.5 & 2 & 2.5 & 3 & 3.5 \\ \hline y & 1.817 & 1.935 & 2.080 & 2.241 & 2.410 & 2.584 \end{array}$
(a) Using the trapezium rule with all the values of $y$ in the given table, obtain an estimate for

$\int_{1}^{3.5} \sqrt[3]{x^2 + 5}\,\mathrm{d}x$

giving your answer to 3 significant figures.

(3 marks)

Using your answer to part (a) and making your method clear, estimate
(b) (i) $\displaystyle\int_{1}^{3.5} \left(4 + \sqrt[3]{x^2 + 5}\right)\mathrm{d}x$

(ii) $\displaystyle\int_{1}^{3.5} \sqrt[3]{8x^2 + 40}\,\mathrm{d}x$

(3 marks)
Level 3 · 6 marksTry it
2
A continuous curve has equation $y = \log_3(2x + 5)$

The table below gives values of $y$ for this curve, each to 4 decimal places where appropriate.

$\begin{array}{c|cccccc} x & 0 & 0.4 & 0.8 & 1.2 & 1.6 & 2 \\ \hline y & 1.4650 & 1.6001 & 1.7177 & 1.8218 & 1.9153 & 2 \end{array}$

Using the trapezium rule with all the values of $y$ in the given table,
(a) find an estimate for

$\int_{0}^{2} \log_3(2x + 5)\,\mathrm{d}x$

giving your answer to 3 decimal places.

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

(i) $\displaystyle\int_{0}^{2} \log_3\sqrt{2x + 5}\,\mathrm{d}x$

(ii) $\displaystyle\int_{0}^{2} \log_3(18x + 45)\,\mathrm{d}x$

(2 marks)
Level 3 · 5 marksTry it
3
Use the trapezium rule with 4 strips to estimate $\displaystyle\int_0^2 \sqrt{1 + x^3}\,dx$ , giving your answer to 3 decimal places.
Level 3 · 3 marksTry it
4
(a) Use the trapezium rule with 4 strips to estimate $\displaystyle\int_0^1 e^{x^2}\,dx$ to 3 decimal places.
(b) State, with a reason, whether this is an over- or under-estimate.
Level 3 · 5 marksTry it
5
(a) Use the trapezium rule with 4 strips to estimate $\displaystyle\int_1^5 \dfrac1x\,dx$ to 4 decimal places.
(b) The exact value is $\ln 5 = 1.6094$ . Explain why the estimate is larger.
Level 3 · 5 marksTry it

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