Numerical Integration in MATLAB

Introduction

Numerical integration, also known as numerical quadrature, is a method of approximating definite integrals of functions when an analytical solution is not available. In this guide, we'll introduce you to numerical integration in MATLAB with sample code and examples.

Defining the Function

First, you need to define the function that you want to integrate numerically. In this example, we'll use a simple function \(f(x) = x^2\).

% Example: Define the function to integrate
f = @(x) x^2;
    

Choosing an Integration Method

MATLAB offers various numerical integration methods, including the trapezoidal rule, Simpson's rule, and Gaussian quadrature. You can choose the appropriate method based on the characteristics of your function and the desired accuracy. In this guide, we'll use the trapezoidal rule.

% Example: Choose the integration method
integration_method = 'trapezoidal';
    

Performing Numerical Integration

You can perform numerical integration using MATLAB's built-in functions, such as trapz for the trapezoidal rule. Here's how to calculate the definite integral of the function over a specified interval \([a, b]\):

% Example: Perform numerical integration
a = 0;  % Lower limit of integration
b = 1;  % Upper limit of integration
integral_result = trapz(linspace(a, b, 1000), arrayfun(f, linspace(a, b, 1000)));
    

Displaying the Result

You can display the result of the numerical integration, which will give you an approximation of the definite integral of the function over the specified interval.

% Example: Display the integration result
disp(['Definite integral of f(x) from ', num2str(a), ' to ', num2str(b), ' is approximately ', num2str(integral_result)]);
    

Conclusion

This guide has introduced you to numerical integration in MATLAB. Numerical integration is a valuable technique when you need to approximate integrals for functions without analytical solutions. As you gain experience, you can explore more advanced integration methods and use them in various mathematical and scientific applications.

Enjoy performing numerical integration in MATLAB for your integration needs!