MATH 132 -- Calculus for Physical and Life Science 2
Approximate Integration in Maple
February 13, 2008
Let's consider the definite integral 
This is one where we can
find a symbolic indefinite integral to apply the Evaluation Theorem. Here's the
way it's done with Maple (the form is different from, though equivalent to,
what our tables would say).
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(1) |
The definite integral, in exact form (val) and then in a decimal approximation (ExactValue):
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(2) |
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(3) |
Now we compute some approximations using the Left, Right, and Midpoint Riemann
sums, the Trapezoidal rule, and Simpson's Rule, together with the error (the difference
ExactValue - ApproximateValue) for each method:
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(4) |
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(5) |
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(6) |
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(7) |
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(8) |
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(9) |
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(10) |
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(11) |
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(12) |
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(13) |
Some questions about what we are seeing here:
Which of the estimates are underestimates? And how can you tell?
2. Which of the estimates are overestimates? How can you tell?
3. Which method is "better" here -- Trapezoidal Rule or Midpoint Riemann Sum?
4. Which is the "best" method overall?
Note that the weighted average
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(14) |
gives the same value as Simpson's Rule (!?) Is this a coincidence?