The Trapezium Rule

The trapezium rule is a way of estimating the area under a curve. We know that the area under a curve is given by integration, so the trapezium rule gives a method of estimating integrals. This is useful when we come across integrals that we don't know how to evaluate.

The trapezium rule works by splitting the area under a curve into a number of trapeziums, which we know the area of.

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If we want to find the area under a curve between the points x0 and xn, we divide this interval up into smaller intervals, each of which has length h (see diagram above).

Then we find that:

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where y0 = f(x0) and y1 = f(x1) etc

If the original interval was split up into n smaller intervals, then h is given by: h = (xn - x0)/n

Example

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You may be also be asked in your exam to summarise an advantage and a limitation of the trapezium rule.

An Advantage: The trapezium rule is relatively simple to apply and understand compared to some other numerical integration methods.

A Limitation: The trapezium rule may not provide accurate results for functions with rapidly changing slopes or high curvature, requiring more intervals to achieve accuracy, which can increase computational complexity.