## The Trapezium Rule

The trapezium rule is a numerical method for estimating integrals. It is most useful when there is no analytical answer to an integral, and only a number is needed. It works by approximating the area under the curve by a series of trapezia, then evaluating the areas and adding them up.

The area under the curve is approximated by a series of trapezia.

The formula for the integral rule iswhereis the step size or the increment by which the values ofincreases, 1 in the diagram above andis defined byeg

For ease of calculation it is a good idea to tabulate the values of the

 i 0 1 2 3 4 5 6 xi 0 1 2 3 4 5 6 yi 0 5 8 9 8 5 0

This estimate just calculated is an underestimate for the integral. The curve is concave – it curves down. In general the trapezium method gives an underesimate for concave integrals and an overestimate for convex integrals, an example of which is given below.

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