Monte Carlo Integration GUI

A comprehensive MATLAB graphical user interface for numerical integration using Monte Carlo methods. This tool provides an intuitive way to compute single and double integrals with real-time visualization and error analysis.

Single Integral Integration

Compute 1D integrals with real-time function visualization and statistical analysis

Double Integral Integration

Compute 2D integrals with stunning 3D surface plots, contour maps, and comprehensive analysis

---

Table of Contents

• Features
• Installation
• Usage
• Examples
• Requirements

---

What is Monte Carlo Integration?

Monte Carlo integration is a numerical method that uses random sampling to estimate integrals.

Instead of calculating the exact area under a curve, the method works by:

1. Generating random points in the integration region.
2. Evaluating the function at those points.
3. Averaging the results to approximate the mean value of the function.
4. Scaling by the region size (interval length, rectangle area, etc.) to estimate the integral.

• The more samples you use, the closer the estimate gets to the true value.
• This method is especially powerful for high-dimensional problems where traditional techniques are inefficient.

---

Features

Core Functionality

• Single Integral (1D): Compute ∫f(x)dx over [a,b]
• Double Integral (2D): Compute ∫∫f(x,y)dxdy over rectangular domains
• Real-time Visualization: Interactive plots showing function behavior and sampling
• Statistical Analysis: Automatic error estimation and confidence intervals

Advanced Analytics

• Monte Carlo Sampling: Efficient random sampling with user-defined sample sizes
• Error Analysis: Standard error calculation and 95% confidence intervals
• Convergence Visualization: Multiple plot types for comprehensive analysis

Installation

Method 1: Direct Download

1. Download the MonteCarloIntegrationGUI.m file
2. Place it in your MATLAB working directory
3. Run the function:

   MonteCarloIntegrationGUI()

Method 2: Clone Repository

git clone https://github.com/yourusername/monte-carlo-integration-gui.git
cd monte-carlo-integration-gui

Then in MATLAB:

MonteCarloIntegrationGUI()

Usage

Getting Started

1. Launch the GUI: MonteCarloIntegrationGUI()
2. Choose a tab: Single Integral or Double Integral
3. Enter your function using MATLAB syntax
4. Set integration limits and sample size
5. Click Calculate to see results and visualizations

Function Syntax

• Single Integral: @(x) x.^2 + sin(x)
• Double Integral: @(x,y) x.^2 + y.^2 + x.*y

Sample Sizes

• Quick Test: 10,000 samples
• Standard: 100,000 samples (recommended)
• High Precision: 1,000,000+ samples

Examples

Single Integral Examples

Polynomial Function

Function: @(x) x.^3 + 2*x.^2 - x + 1
Domain: [0, 2]
Expected Result: ~8.67

Gaussian Function

Function: @(x) exp(-(x-2).^2) .* sqrt(x)
Domain: [0, 5]
Expected Result: Beautiful bell curve visualization

Double Integral Examples

Simple Polynomial

Function: @(x,y) x.^2 + y.^2
Domain: [0,1] × [0,1]
Expected Result: 0.6667 (exact: 2/3)

3D Gaussian Hill Most Spectacular

Function: @(x,y) exp(-(x.^2 + y.^2))
Domain: [-2,2] × [-2,2]
Expected Result: ~3.14159 (≈ π)

Requirements

System Requirements

• MATLAB: R2019b or later (App Designer required)
• Toolboxes: Statistics and Machine Learning Toolbox (recommended)
• RAM: 4GB+ recommended for large sample sizes
• Display: 1200×800 minimum resolution

MATLAB Functions Used

• uifigure, uitabgroup, uipanel (App Designer)
• rand, mean, std (Core functions)
• surf, contourf, scatter (Visualization)
• histogram, plot, colorbar (Graphics)

References

1. Numerical Recipes: Press, W. H., et al. Numerical Recipes in C
2. Monte Carlo Methods: Rubinstein, R. Y. Simulation and the Monte Carlo Method
3. MATLAB Documentation: MathWorks App Designer Guide