Update README.md

README.md

 # openMP-HW
 
-This repository contains two parallel C/C++ programs developed using OpenMP for numerical integration and area approximation tasks.
\ No newline at end of file
+This repository contains two parallel C/C++ programs developed using OpenMP for numerical integration and area approximation tasks.
+
+---
+
+## Overview
+This homework includes two C/C++ programs that leverage OpenMP for parallel computation:
+1. **Numerical Integration** of \( f(x) = \frac{\ln(x)}{x} \) over the interval \([1, 10]\).
+2. **Mandelbrot Set Area Approximation** using the Monte Carlo method.
+
+Both programs contain sequential and parallel implementations to analyze performance and efficiency using OpenMP, along with an automated build and run process via Make.
+
+---
+
+### Question 1: Numerical Integration
+**Objective**: Calculate the integral of \( \frac{\ln(x)}{x} \) over \([1, 10]\) using a numerical integration technique that partitions the area under the curve into rectangles with equal width. The sum of these rectangles provides an approximation of the integral.
+
+- **Approach**: Implemented the Riemann sum (rectangle) method, dividing the interval into `num_rectangles` of equal width.
+- **Parallelization**: Used OpenMP to distribute rectangle area calculations across multiple threads. 
+- **Analysis**: Compared the execution time of the sequential vs. parallel versions to understand the impact of parallelization on performance.
+
+### Question 2: Mandelbrot Set Area Approximation
+**Objective**: Approximate the area of the Mandelbrot set using the Monte Carlo method. Points are randomly generated within a defined region in the complex plane, and those that remain bounded after iterative calculations are counted to estimate the area.
+
+- **Approach**: Randomly generated points in the complex plane, checking if each point belongs to the Mandelbrot set by iterating a complex function up to a set number of iterations.
+- **Parallelization**: OpenMP was used to parallelize the point-checking process across multiple threads.
+- **Analysis**: Compared execution time of sequential vs. parallel implementations and analyzed factors affecting parallel efficiency.
+
+---
+
+## Files (To Do)
+- **`integral.c`**: Program for parallel numerical integration.
+- **`mandelbrot.cpp`**: Program for parallel area approximation of the Mandelbrot set using Monte Carlo.
+- **`Makefile`**: Automates building, running, and cleaning both programs.
+
+---
+
+## Compilation and Execution
+
+### Prerequisites
+- **GCC with OpenMP support**: Ensure `gcc` and `make` are installed.
+
+### Steps
+1. **Build**: To compile both programs, run:
+   ```bash
+   make buid_all
+   ```
+2. **Run Numerical Integration Program**:
+   ```bash
+   make run Pr=integral
+   ```
+3. **Run Mandelbrot Area Program**:
+   ```bash
+   make run Pr=mandelbrot
+   ```
+4. **Profiling the Program Executing**:
+   ```bash
+   make Profile Pr=mandelbrot
+   ```
+
+5. **Clean**: To remove compiled binaries, run:
+   ```bash
+   make clean
+   ```
+
+Each program outputs the calculated area and the execution time for both sequential and parallel versions, allowing comparison of their performance.
+
+---
+
+## Results Analysis
+
+- **Integration Program**:
+  - **Expected Output**:
+  - **Performance**: 
+
+- **Mandelbrot Area Program**:
+  - **Expected Output**: 
+  - **Performance**:
+
+---
+
+## Conclusion
+This homework illustrates the effectiveness of OpenMP in parallelizing computational tasks.
+Performance comparisons between sequential and parallel implementations reveal insights into how thread distribution and workload balance influence speed and efficiency.
+
+---
+