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
Use the trapezium rule with 4 strips to estimate $\displaystyle\int_0^4 \sqrt{x^2 + 1}\,dx$ , giving your answer to 3 decimal places.
Level 2 · 3 marksTry it
2
Use the trapezium rule with 5 ordinates to estimate $\displaystyle\int_1^3 2^x\,dx$ , giving your answer to 3 decimal places.
Level 2 · 3 marksTry it
3
The table below shows corresponding values of $x$ and $y$ for $y = \sqrt{5 + x^4}$

The values of $y$ are given to 3 decimal places as appropriate.

$\begin{array}{c|ccccc} x & 0 & 0.4 & 0.8 & 1.2 & 1.6 \\ \hline y & 2.236 & 2.242 & 2.326 & 2.660 & 3.399 \end{array}$
(a) Use the trapezium rule with all the values of $y$ in the table to find an estimate for

$\int_0^{1.6} \sqrt{5 + x^4}\,dx$

giving your answer to 3 significant figures.

(3 marks)
(b) Use your answer to part (a) to find an estimate for

$\int_0^{1.6} \left(1 + 4\sqrt{5 + x^4}\right)dx$

(2 marks)
Level 2 · 5 marksTry it
4
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
5
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

23 more questions on the trapezium rule

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