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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) – PowerPoint PPT presentation

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Title: Improper integrals


1
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

2
Case 1 Unbounded integrals
3
Case 1, continued integrals over whole line
4
Case 2 Integrand blows up
5
More Examples
6
Comparison principle
  • Can often show convergence or divergence of a
    complicated integral by comparing it to a simpler
    one

7
Model examples
8
Numerical Integration
  • Some integrals are impossible to integrate
    exactly
  • Instead find approximate values using various
    numerical integration techniques
  • Example

9
Midpoint Rule
  • Divide a,b into n subintervals of equal length,
    x0 , x1 , ., xn
  • Then

10
Visualization of midpoint rule
11
Trapezoidal Rule
  • Partition a,b into n intervals of equal length,
    with endpoints x0 , ,xn

12
Visualization of the trapezoid rule
13
Another Method
  • Simpsons rule Sn (Tn 2Mn)/3
  • Click here to see various rules in action
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