How Monte Carlo Integration Quickly Estimates Double Integrals

1 Definite Integral Calculation

Calculating definite integrals is the entry point of Monte Carlo methods in computational mathematics. For many complex problems requiring multiple integrals, Monte Carlo can provide fast, low‑precision approximations, and its error does not depend on the dimensionality, making it superior to uniform grid formulas.

1.1 Problem 1

Compute a double integral.

1.2 Solution

According to the geometric meaning of the integral, it represents the volume of a solid whose top surface is defined by the function and bottom by the plane, enclosed in a rectangular box of volume 4. By generating uniformly distributed points inside the box and counting those that fall inside the target region, the ratio approximates the probability, which equals the desired volume divided by 4, yielding an estimate of the integral.

Python code implementing the Monte Carlo estimation:

<code>from numpy.random import uniform
import numpy as np
N = 10000000
x = uniform(-1, 1, size=N)
y = uniform(-1, 1, N)
z = uniform(0, 1, N)
n = np.sum((x**2 + y**2 <= 1) & (z >= 0) & (z <= np.sqrt(1 - x**2)))
I = n / N * 4
print("I的近似值为：", I)
</code>

The program outputs an approximate value I ≈ 2.6670568.

Source: Si Shougui, Sun Xijing. Python Mathematics Experiments and Modeling (2020). Science Press.