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NUMERICAL INTEGRATION. Motivation: Most such integrals cannot be evaluated explicitly. Many others it is often faster to integrate them numerically rather than evaluating them exactly using a complicated antiderivative of f(x) Example:

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numerical integration
NUMERICAL INTEGRATION
  • Motivation:

Most such integrals cannot be evaluated explicitly. Many others it is often faster to integrate them numerically rather than evaluating them exactly using a complicated antiderivative of f(x)

  • Example:

The solution of this integral equation with Matlab is 1/2*2^(1/2)*pi^(1/2)*FresnelS(2^(1/2)/pi^(1/2)*x)

we cannot find this solution analytically by techniques in calculus.

course content
Course content
  • Methods of Numerical Integration
    • Trapezoidal Rule’s
    • 1/3 Simpson’s method
    • 3/8 Simpson’s method
  • Applied in two dimensional domain
slide4
Function f approximately by function fp. Then,

where fp is a linear polynomial interpolation, that is

By substitution u=x-x0we have

where

slide6
For two interval, we can use summation operation to derive the formula of two interval trapezoidal that is

where

slide8
Similar to two interval trapezoidal, we can derive three interval trapezoidal formula that is

where

  • Thus, for n interval we have

where and

for

slide10
Function f approximately by function fp. Then,

where fp is a quadratic polynomial interpolation, that is

By substitution u=x-x0we have

where

slide12
For 4 subinterval we have

where

  • Thus, for n subinterval we have

where and

slide14
Similar to 1/3 Simpson’s method, f approximately by function fp where fp is a cubic polynomial interpolation, that is

By substitution u=x-x0we have

where and

slide16
A double integration in the domain is written as
  • The numerical integration of above equation is to reduce to a combination of one-dimensional problems
slide17
Procedure:
  • Step 1: Define

So, the solution is

  • Step 2: Divided the range of integration [a,b] into

N equispaced intervals with the interval size

So, the grid points will be denoted by

and then we have

slide18
Step 3: Divided the domain of integration

into N equispaced intervals with the interval size

So, the grid points denoted by

  • Step 4: By Applying numerical integration for one-dimensional (for example the trapezoidal rule) we have

for

slide19
Step 5: By applying numerical integration (for example trapezoidal rule) in one-dimensional domain we have the solution of double integration is