Calculate Area Under The Curve in Matlab Using Integral

The integral function in MATLAB calculates the area under a curve defined by a function handle or anonymous function. This guide explains how to use integral to compute definite integrals, including syntax, examples, and best practices.

Introduction

The integral function in MATLAB is a powerful tool for numerical integration. It calculates the area under a curve between specified limits, which is essential in many scientific and engineering applications.

MATLAB's integral function uses adaptive quadrature algorithms to provide accurate results for a wide range of functions.

Basic Syntax

The basic syntax for the integral function is:

Q = integral(fun, a, b)

Where:

fun is a function handle or anonymous function
a is the lower limit of integration
b is the upper limit of integration
Q is the computed integral value

For example, to compute the integral of sin(x) from 0 to π:

Q = integral(@sin, 0, pi)

Examples

Example 1: Simple Integral

Calculate the integral of x² from 0 to 1:

Q = integral(@(x) x.^2, 0, 1)

Result: 0.3333 (1/3)

Example 2: Multiple Variables

Calculate the integral of x*y from 0 to 1 for both x and y:

Q = integral2(@(x,y) x.*y, 0, 1, 0, 1)

Result: 0.25

Example 3: Using Options

Calculate the integral with specified options:

opts = optimoptions('integral', 'AbsTol', 1e-8, 'RelTol', 1e-6);
Q = integral(@(x) exp(-x.^2), -Inf, Inf, opts);

Result: 1.7725 (approximation of √π)

Advanced Usage

For more complex integrals, you can use:

Anonymous functions for multi-variable integrals
Options to control accuracy and algorithm selection
Vectorized functions for improved performance

When working with infinite limits, MATLAB uses adaptive algorithms to handle the integration properly.

Troubleshooting

Common issues and solutions:

Slow computation: Use vectorized functions and adjust options for better performance.
Accuracy problems: Increase the absolute and relative tolerances.
Singularities: Use appropriate limits or options to handle singular points.

FAQ

What is the difference between integral and quad?

The integral function uses adaptive quadrature algorithms and is generally more accurate and flexible than quad, which uses fixed quadrature rules.

Can integral handle complex functions?

Yes, integral can handle complex-valued functions as long as they are properly defined.

How do I handle infinite limits?

Use -Inf or Inf as the limits, and MATLAB will use adaptive algorithms to handle the integration properly.