## 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 ofincrease, 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 |

x-i | 0 | 1 | 2 | 3 | 4 | 5 | 6 |

y-i | 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.