 # 1 NUMERICAL INTEGRATION Motivation: Most such integrals cannot be evaluated explicitly. Many others it is often faster to integrate them numerically rather.

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1 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.

2 Methods of Numerical Integration –Trapezoidal Rule’s –1/3 Simpson’s method –3/8 Simpson’s method Applied in two dimensional domain Course content

3 Trapezoidal Rule’s f fpfp

4 Function f approximately by function fp. Then, where fp is a linear polynomial interpolation, that is By substitution u=x-x 0 we have where

5 Trapezoidal Rule’s f fpfp

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

7 Trapezoidal Rule’s f fpfp

8 Similar to two interval trapezoidal, we can derive three interval trapezoidal formula that is where Thus, for n interval we have whereand for

9 1/3 Simpson’s f fpfp

10 Function f approximately by function fp. Then, where fp is a quadratic polynomial interpolation, that is By substitution u=x-x 0 we have where

11 f fpfp 1/3 Simpson’s

12 For 4 subinterval we have where Thus, for n subinterval we have whereand

13 3/8 Simpson’s f fpfp

14 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-x 0 we have where and

15 Numerical Integration in a Two Dimensional Domain c(x) d(x) =a b=

16 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

17 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

18 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

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

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