Numerical Integration

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