Answer:
Monte Carlo integration is a numerical method for approximating the value of an integral using random sampling. To use Monte Carlo integration in this case , we can approximate the value of the integral by taking the average value of the function over the given area, weighted by the area of the rectangle. This can be expressed as:
integral f(x,y) dA = approximate integral f(x,y) dA approximate integral f(x,y) dA = (total area of rectangle) * average(f(x,y))
We can use the 15 points given to estimate the average value of the function over the given area by evaluating the function at each point and taking the mean. To convert each point to the appropriate range, we need to map the interval (0,1) to the interval (1,8) for x and (1,5) for y. This can be done using the following formulas:
x = a + (b-a) * u y = c + (d-c) * v
where a=1, b=8, c=1, d=5, and u and v are the random numbers generated from the pseudo-random number generator.
Here's the code to implement this:
import numpy as np
# Define the function to be integrated
def f(x, y):
return x * y * np.log(x+y) + 7
# Define the corners of the rectangle
a, b, c, d = 1, 8, 1, 5
# Define the 15 points
points = np.array([[0.14581, 0.62102], [0.04793, 0.38346], [0.96691, 0.50057],
[0.61175, 0.83935], [0.03211, 0.66880], [0.71623, 0.71778],
[0.15910, 0.01757], [0.53173, 0.33055], [0.05475, 0.46542],
[0.73619, 0.70010], [0.15362, 0.77275], [0.18846, 0.50957],
[0.56782, 0.19728], [0.59664, 0.09514], [0.36417, 0.46100]])
# Map the points to the
Step-by-step explanation: