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2 Chapter 6. Quadrature Figure 6.1. Quadrature. can recursively apply the additive property to each of the intervals [ a,c ] and [ c,b ]. The resulting algorithm will adapt to the integrand automatically, partitioning the interval into subintervals with ﬁne spacing where the integrand is varying rapidly and coarse spacing where the integrand is varying slowly. 6.2 Basic Quadrature Rules The derivation of the quadrature rule used by our Matlab function begins with two of the basic quadrature rules shown in Figure 6.2: the midpoint rule and the trapezoid rule . Let h = b - a be the length of the interval. The midpoint rule, M , approximates the integral by the area of a rectangle whose base has length h and whose height is the value of the integrand at the midpoint: M = hf ± a + b 2 . The trapezoid rule, T , approximates the integral by the area of a trapezoid with base h and sides equal to the values of the integrand at the two endpoints: T = h f ( a ) + f ( b ) 2 . The accuracy of a quadrature rule can be predicted in part by examining its behavior on polynomials. The order of a quadrature rule is the degree of the lowest degree polynomial that the rule does not integrate exactly. If a quadrature rule of order p is used to integrate a smooth function over a small interval of length h , then a Taylor series analysis shows that the error is proportional to h p . The midpoint
6.2. Basic Quadrature Rules 3 Midpoint rule

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