Numerical integration;Gaussian integration one point, two point and three point method.

ppt on Numerical integration and Gaussian integration one point, two point and three point method.

1. Numerical Integration Gaussian integration one point, two point and three point method.

2. What is Integration ? The process of measuring the area under a curve . Where: f(x) is the integrand a= lower limit of integration b = upper limit of integration

3. Gaussian quadrature In numerical analysis, a  quadrature rule  is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration . An  n -point  Gaussian quadrature rule is a quadrature rule constructed to yield an exact result for polynomials of degree 2 n  − 1 or less by a suitable choice of the points  x i  and weights  w i  for  i  = 1, ...,  n . The domain of integration for such a rule is conventionally taken as [−1, 1], so the rule is stated as

4. One-Point Gaussian Quadrature Rule Consider a function f(x) over interval [-1,1] with sampling point The point one formula is The formula of one point Gaussian quadrature rule,

5. Two-Point Gaussian Quadrature Rule Consider a function f(x) over interval [-1,1] with sampling point and The two point formula is, The formula of one point Gaussian quadrature rule,

6. Three-Point Gaussian Quadrature Rule Consider a function f(x) over interval [-1,1] with sampling point and The two point formula is , The formula of Three point Gaussian quadrature rule,

7. Example 1 Evaluate by one point , two point & Three point Gaussian quadrature. Here, Using one point method Using Two point Method

8. Three point method

9. Example 2 Evaluate by one point ,two point ,three point Gaussian formula. Let where a=0,b=1 now

10. t=0 One point Method t=1

11. Two point method Three point method

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