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The trapezium rule estimates the area under a curve that cannot be found exactly by dividing it into strips and approximating each with a trapezium. This topic covers applying it and whether it over- or under-estimates. It's a key topic in A-Level and International A-Level (iAL) Pure Mathematics. This free interactive eClassroom course walks through each step with worked examples and instant-feedback questions, so you can practise until it's secure.
How you do it
Divide the interval into equal strips and find the height at each boundary. The estimate is half the strip width times the first height plus the last, plus twice everything in between. Whether it over or underestimates depends on whether the curve bends up or down.
Before you start
This builds directly on substitution and area. Later work on numerical methods assumes it. It belongs to the Pure 2 specification.
What you learn
The trapezium rule estimates a definite integral by slicing the area into strips and treating each as a trapezium. and mixed problem-solving. The rule works for any function — including surds, where an exact answer is often awkward. The trapezium rule is essential for functions with no elementary antiderivative — or where an exact integral is beyond the current toolkit. Each strip's top is a straight chord joining two points on the curve. The course works through the trapezium rule & a table of ordinates, strips vs ordinates; surd functions, trig, exponential & log functions, over- or under-estimate? Every section has instant-feedback practice questions matched to Edexcel International A-Level and 9MA0 Pure, and the course finishes with full exam-style questions you mark yourself.
Related Topics
Trapezium Rule is part of the Integration strand of our free A-level maths courses, studied in the AS year. There are 5 teaching sections and 5 fully worked examples, plus 5 sets of check questions. It sits alongside Volumes of Revolution, Area Between Two Curves and Area Under a Curve in the same strand. The Integration revision engine gives you as many extra questions as you want, tracked on your skill tree.
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