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Numerical Integration

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Numerical Integration Approximating Definite Integral The Trapezoidal Rule Some elementary functions do not have antiderivatives that are elementary functions. – PowerPoint PPT presentation

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Title: 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
Simpsons Rule
  • The number of parabolas must be even. Simpsons
    will not work with an odd number.

7
Simpsons Rule
  • Let f be continuous on a, b. Simpsons Rule for
    approximating

8
Simpsons Rule
9
Approximation with Simpsons Rule
10
Simpsons 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 Simpsons 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
x 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
y 4.32 4.36 4.58 5.79 6.14 7.25 7.64 8.08 8.14
14
Using a Table
  • The definite integral we are looking for is

15
Trapezoidal Rule
16
Simpsons Rule
17
Your Turn
  • Do p. 305 problems 1 19 odd 33, 35 and 43
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