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**Numerical Integration Lecture (I)1**

Chapter 8 Numerical Integration Lecture (I)1 1 Ref.: “Applied Numerical Methods with MATLAB for Engineers and Scientists”, Steven Chapra, 2nd ed., Ch. 17, McGraw Hill, 2008. Dr. Jie Zou PHY3320

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Outline Introduction What is integration? When do we need numerical integration? Applications of integration in engineering and science Newton-cotes formulas (1) The trapezoidal rule Error of the Trapezoidal rule The composite trapezoidal rule Implementation in MATLAB Dr. Jie Zou PHY3320

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**Introduction What is integration?**

Mathematically: A definite integration is represented by It means: The total value, or summation, of f(x)dx over the range x = a to b. Graphical representation: For functions lying above the x axis, the integral corresponds to the area under the curve of f(x) between x = a and b. When do we need numerical integration (also referred to as quadrature)? Functions that are difficult to or cannot be integrated analytically. Only a table of discrete data are available. Ref. Fig Graphical representation of the integral Dr. Jie Zou PHY3320

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**Applications of integration in engineering and science**

Examples related to “the integral as the area under a curve”: Examples related to the analogy between integration and summation: An example: To determine the mean of a continuous function Ref. Fig. 17.3

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**Newton-cotes formulas**

Basic strategy: Replacing a complicated function or tabulated data with a polynomial that is easy to integrate. fn(x) = a0 + a1x + … + an-1xn-1+anxn n: The order of the polynomial. Ref. Fig The approx. of an integral by the area under (a) a straight line and (b) a parabola Ref. Fig The approx. of an integral by the area under three straight-line segments Dr. Jie Zou PHY3320

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**Newton-cotes formulas: (1) The trapezoidal rule**

Basic idea: Replacing the complicated function or tabulated data with a polynomial or a series of polynomials of the first order (linear). Single and Composite applications Single application formula: Composite application formula: Ref. Fig Single application Ref. Fig Composite application

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**Error of the trapezoidal rule**

For single applications, an estimate for the error: If the function being integrated is linear, Et = 0; otherwise, Et 0. For composite applications, an estimate for the error: If the number of segments is doubled, Et is approximately quartered. Here, Ref. Fig Truncation error for a single application of the trapezoidal rule

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**Example: Composite application of the trapezoidal rule**

Example 17.2 (Ref.): Use the two-segment trapezoidal rule to estimate the integral of f(x) = x – 200x x3 – 900x x5 from a = 0 to b = 0.8. Also, find the true error Et and the approximate error, Ea. (1) By hand. (2) Implement on a computer-write an M-file. x0 = a x1 x2 = b Two segments n = 2; Dr. Jie Zou PHY3320

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Results n h I t (%) 2 0.4 1.0688 34.9 3 0.2667 1.3695 16.5 4 0.2 1.4848 9.5 5 0.16 1.5399 6.1 6 0.1333 1.5703 4.3 7 0.1143 1.5887 3.2 8 0.1 1.6008 2.4 9 0.0889 1.6091 1.9 10 0.08 1.6150 1.6 Dr. Jie Zou PHY3320

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**Implementation of composite trapezoidal rule on a computer**

Write an M-file called My_Trapezoidal_Rule.m to do Example 17.2. A copy of the code will be handed out later. Dr. Jie Zou PHY3320

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