Maths A Level revision: The trapezium rule

The trapezium rule, one of A Level math’s more bizarre, weird and unfortunately time consuming subject areas.

The trapezium rule provides you with a way to estimate the value of an integral you can’t do. It involves splitting the area under the under up into trapeziums which are then totalled to give an estimate for the area.

Trapezium rule

If we wanted to find the region between the curve and x = 7 and x = 12 we would split it up into equal width strips, with more strips leading to a more accurate result. We’d then find the y values for the x values of our strips from either the graph or the equation if we are given it. We then use the trapezium rule equation:

A \approx \frac{h}{2}[y_0 + y_n + 2(y_1 + y_2 + ... + y_{n-1})]

Where n is the number of strips and h is the width of the strips.

Example

Example question: Use the trapezium rule with 4 strips to estimate the area enclosed by the curve of \frac{x^2 + 2x}{(1 - x^4)(x - 1)} between x = 2 and x = 6

Worked solution: For 4 strips between x = 2 and x = 6 each strip needs to have a width of 1, so our x values are 2, 3, 4, 5 and 6 (note that there are n + 1 x values). It helps to draw up a table to calculate the corresponding y values from the equation:

xy
2-0.53
3-0.094
4-0.031
5-0.014
60.0074

The minus numbers here simply indicate that the area is below the x-axis.

We can now estimate the area:

A \approx \frac{1}{2}[-0.53 + -0.0074 + 2(-0.094 + -0.031 + -0.014)]

A \approx \frac{1}{2}(-0.5374 + -0.278) = \frac{-0.8154}{2} = -0.4077

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