In mathematics, an approximate method for finding the value of an integral. Regarding the integral as the area of a region between a curve and the x-axis, this region is divided into n trapezia, width h, each of which has area ½h(yr + yr + 1). The area of the whole region is ½h[y0 + 2(y1 + y2 + ...) + yn].
In mathematics, the trapezium rule (the British term) or trapezoid rule (the American term) is a way to approximately calculate the definite integral
The trapezium rule works by approximating the region under the graph of the function f(x) by a trapezium and calculating its area. It follows that
To calculate this integral more accurately, one first splits the interval of integration [a,b] into n smaller subintervals, and then applies the trapezium rule on each of them. One obtains the composite trapezium rule:
This can alternatively be written as:
The trapezium rule is one of a family of formulas for numerical integration called Newton-Cotes formulas. Simpson's rule and other like methods can be expected to improve on the trapezium rule for functions which are twice continuously differentiable; Moreover, the trapezium rule tends to become extremely accurate when periodic functions are integrated over their periods, a fact best understood in connection with the Euler-Maclaurin summation formula.
An advantage of the trapezium rule is that the sign of the error of the approximation is easily known.
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