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Numerical Integration Approximating Definite Integral

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The Trapezoidal Rule Some elementary functions do not have antiderivatives that are elementary functions. One way to approximate the definite integral is to use trapezoids. This is more accurate than using rectangles because there is less extra space included or not included. (see figure 4.43 p. 301)

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The Trapezoidal Rule Let f be continuous on [a, b]. The Trapezoidal Rule for approximating

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The Trapezoidal Rule Use the trapezoidal rule to approximate

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The Trapezoidal Rule

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Simpson’s Rule The number of parabolas must be even. Simpson’s will not work with an odd number.

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Simpson’s Rule Let f be continuous on [a, b]. Simpson’s Rule for approximating

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Simpson’s Rule

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Approximation with Simpson’s Rule

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Simpson’s Rule

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Using a Table The table lists several measurements gathered in an experiment to approximate an unknown continuous function y = f(x). Approximate the integral using the Trapezoidal Rule and Simpson’s Rule.

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Using a Table When given a table, the values for f(x) are already determined for you. A table makes this process much easier.

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Using a Table x0.000.250.500.751.001.251.501.752.00 y4.324.364.585.796.147.257.648.088.14

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Using a Table The definite integral we are looking for is

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Trapezoidal Rule

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Simpson’s Rule

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Your Turn Do p. 305 problems 1 – 19 odd; 33, 35 and 43

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