Monte carlo simulation integration

Question

I'm trying to see how each point in a monte carlo simulation can be seen if it is in the area that is graphed. What is a good method for this?

# Definition of monte carlo operation
def montecarlo(func, x1, y1, x2, y2):
    # Define the rectangle for integration
    # rectangle = (x2 - x1) * (y2 - y1)

    coordinates_list = []
    list_random_x_value = []
    list_random_y_value = []

    # Generate random points in the rectangle
    for i in range(0, 100):
        random_x_value = random.uniform(x1, x2)
        random_y_value = random.uniform(y1, y2)
        list_random_x_value.append(random_x_value)
        list_random_y_value.append(random_y_value)
        # coordinates = [random_x_value, random_y_value]
        # coordinates_list.append(coordinates)
        # print(coordinates_list)

    list_x_values = []
    list_y_values = []


    # Plot of graph
    for values in np.arange(x1, x2, 0.01):
        list_x_values.append(values)
        y_values = func(values)
        list_y_values.append(y_values)

Answer 1

I think you mean something like this?

import numpy as np
import random
import matplotlib.pyplot as plt

# Definition of monte carlo operation
def montecarlo(func, x1, y1, x2, y2):
    # Define the rectangle for integration
    # rectangle = (x2 - x1) * (y2 - y1)

    coordinates_list = []
    list_random_x_value = []
    list_random_y_value = []

    # Generate random points in the rectangle
    for i in range(0, 100):
        random_x_value = random.uniform(x1, x2)
        random_y_value = random.uniform(y1, y2)
        list_random_x_value.append(random_x_value)
        list_random_y_value.append(random_y_value)
        # coordinates = [random_x_value, random_y_value]
        # coordinates_list.append(coordinates)
        # print(coordinates_list)

    list_x_values = []
    list_y_values = []


    # Plot of graph
    for values in np.arange(x1, x2, 0.01):
        list_x_values.append(values)
        y_values = func(values)
        list_y_values.append(y_values)
        
    plt.plot(list_x_values, list_y_values, c="b")
    plt.scatter(list_random_x_value, list_random_y_value, c="r")
    
montecarlo(np.exp, 0, 0, 2, 1)

However, this is a very crude implementation of MC integration. Nevertheless, here is an easy and somewhat ok guide on how to implement these crude MC methods in python: https://boyangzhao.github.io/posts/monte-carlo-integration