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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

Case 1 Unbounded integrals

Case 1, continued integrals over whole line

Case 2 Integrand blows up

More Examples

Comparison principle

- Can often show convergence or divergence of a

complicated integral by comparing it to a simpler

one

Model examples

Numerical Integration

- Some integrals are impossible to integrate

exactly - Instead find approximate values using various

numerical integration techniques - Example

Midpoint Rule

- Divide a,b into n subintervals of equal length,

x0 , x1 , ., xn - Then

Visualization of midpoint rule

Trapezoidal Rule

- Partition a,b into n intervals of equal length,

with endpoints x0 , ,xn

Visualization of the trapezoid rule

Another Method

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