Improper integrals

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Improper integrals numerical integration

• Improper integrals
• Case 1 Unbounded limits of integration
• Case 2 Integrand blows up (becomes infinite)
• Evaluate improper integrals by taking limits of
  proper integrals
• Basics of numerical integration

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Case 1 Unbounded integrals
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Case 1, continued integrals over whole line
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Case 2 Integrand blows up
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More Examples
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Comparison principle

• Can often show convergence or divergence of a
  complicated integral by comparing it to a simpler
  one

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Model examples
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Numerical Integration

• Some integrals are impossible to integrate
  exactly
• Instead find approximate values using various
  numerical integration techniques
• Example

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Midpoint Rule

• Divide a,b into n subintervals of equal length,
  x0 , x1 , ., xn
• Then

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Visualization of midpoint rule
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Trapezoidal Rule

• Partition a,b into n intervals of equal length,
  with endpoints x0 , ,xn

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Visualization of the trapezoid rule
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Another Method

• Simpsons rule Sn (Tn 2Mn)/3
• Click here to see various rules in action