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

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

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. 17.1 Graphical representation of the integral Dr. Jie Zou PHY3320

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

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. 17.4 The approx. of an integral by the area under (a) a straight line and (b) a parabola Ref. Fig. 17.5 The approx. of an integral by the area under three straight-line segments Dr. Jie Zou PHY3320

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. 17.7 Single application Ref. Fig. 17.9 Composite application

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. 17.8 Truncation error for a single application of the trapezoidal rule

Example: Composite application of the trapezoidal rule Example 17.2 (Ref.): Use the two-segment trapezoidal rule to estimate the integral of f(x) = 0.2 + 25x – 200x2 + 675x3 – 900x4 + 400x5 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

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

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