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Numerical Integration

Numerical Integration. Approximating Definite Integral. 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.

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Numerical Integration

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  1. Numerical Integration Approximating Definite Integral

  2. 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)

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

  4. The Trapezoidal Rule • Use the trapezoidal rule to approximate

  5. The Trapezoidal Rule

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

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

  8. Simpson’s Rule

  9. Approximation with Simpson’s Rule

  10. Simpson’s Rule

  11. 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.

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

  13. Using a Table

  14. Using a Table • The definite integral we are looking for is

  15. Trapezoidal Rule

  16. Simpson’s Rule

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

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