# Improper integrals - PowerPoint PPT Presentation

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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