Numerical Integration

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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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Simpsons Rule

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

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Simpsons Rule

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

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Simpsons Rule
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Approximation with Simpsons Rule
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Simpsons 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 Simpsons 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
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
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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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Simpsons Rule
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Your Turn

• Do p. 305 problems 1 19 odd 33, 35 and 43