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Published byLily Benson Modified over 6 years ago

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Numerical Integration In general, a numerical integration is the approximation of a definite integration by a “weighted” sum of function values at discretized points within the interval of integration.

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Rectangular Rule x=a x=b Approximate the integration,, that is the area under the curve by a series of rectangles as shown. The base of each of these rectangles is x=(b-a)/n and its height can be expressed as f(x i *) where x i * is the midpoint of each rectangle x=x 1 * x=x n * height=f(x 1 *) height=f(x n *) f(x) x

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Trapezoidal Rule x=a x=b x=x 1 x=x n-1 f(x) x The rectangular rule can be made more accurate by using trapezoids to replace the rectangles as shown. A linear approximation of the function locally sometimes work much better than using the averaged value like the rectangular rule does.

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Simpson’s Rule Still, the more accurate integration formula can be achieved by approximating the local curve by a higher order function, such as a quadratic polynomial. This leads to the Simpson’s rule and the formula is given as: It is to be noted that the total number of subdivisions has to be an even number in order for the Simpson’s formula to work properly.

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Examples ixi*xi*f(x i *) 11.1251.42 21.3752.60 31.6254.29 41.8756.59

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Trapezoidal Rule ixixi f(x i ) 11 11.251.95 21.53.38 31.755.36 28 Simpson’s Rule

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