Approximate Integration: The Trapezoidal Rule

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Approximate IntegrationThe Trapezoidal Rule

• Claus Schubert
• May 25, 2000

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Why Approximate Integration?

• Cant always find an antiderivative
• Example

• Dont always know the function

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First Approach Riemann Sums

• Use left or right Riemann sums to approximate the
  integral.

• Left Riemann sum
• Dx length of the n subintervals
• xi endpoints of the subintervals

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Left Riemann Sums
By refining the partition, we obtain better
approximations.
Ln is the sum of all the inscribed rectangles
starting at the left endpoints. It is called a
left endpoint approximation.
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Right Riemann Sums
Rn is the sum of all the inscribed rectangles
starting at the right endpoints. It is called a
right endpoint approximation.
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Left and Right Endpoint Approximations

• Observations

• If Ln underestimates, then Rn overestimates, and
  vice versa

• Approximations get better if we increase n

• Take the average of both approximations

• Idea for improvement

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Trapezoidal Approximation
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Trapezoidal Approximation
Ln
Rn
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Trapezoidal Approximation
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Trapezoidal Approximation
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An Example

• As an example, let us look at .

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An Example
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Error bounds

• Question
• How accurate is the trapezoidal approximation?

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Error bounds An Example

• In our previous example, how large should n be so
  that the error is less than 0.00001 ?

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Error bounds An Example
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Lets Wrap Up

• Approximations are useful if the function cannot
  be integrated or no function is given to begin
  with.

• Left and right endpoint approximations are too
  inaccurate, so take their average.

• The trapezoidal approximation is much more
  accurate than the left/right approximations, but
  better approximations exist (midpoint, Simpsons
  etc.)

• You need a computer to find approximations with
  large n - or you need to get a life!!!