Add Monte Carlo's Integral Approximation (#6235)

src/main/java/com/thealgorithms/randomized/MonteCarloIntegration.java

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+package com.thealgorithms.randomized;
+
+import java.util.Random;
+import java.util.function.Function;
+
+/**
+ * A demonstration of the Monte Carlo integration algorithm in Java.
+ *
+ * <p>This class estimates the value of definite integrals using randomized sampling,
+ * also known as the Monte Carlo method. It is particularly effective for:
+ * <ul>
+ *   <li>Functions that are difficult or impossible to integrate analytically</li>
+ *   <li>High-dimensional integrals where traditional methods are inefficient</li>
+ *   <li>Simulation and probabilistic analysis tasks</li>
+ * </ul>
+ *
+ * <p>The core idea is to sample random points uniformly from the integration domain,
+ * evaluate the function at those points, and compute the scaled average to estimate the integral.
+ *
+ * <p>For a one-dimensional integral over [a, b], the approximation is the function range (b-a),
+ * multiplied by the function average result for a random sample.
+ * See more: <a href="https://en.wikipedia.org/wiki/Monte_Carlo_integration">Monte Carlo Integration</a>
+ *
+ * @author: MuhammadEzzatHBK
+ */
+
+public final class MonteCarloIntegration {
+
+    private MonteCarloIntegration() {
+    }
+
+    /**
+     * Approximates the definite integral of a given function over a specified
+     * interval using the Monte Carlo method with a fixed random seed for
+     * reproducibility.
+     *
+     * @param fx    the function to integrate
+     * @param a     the lower bound of the interval
+     * @param b     the upper bound of the interval
+     * @param n     the number of random samples to use
+     * @param seed  the seed for the random number generator
+     * @return      the approximate value of the integral
+     */
+    public static double approximate(Function<Double, Double> fx, double a, double b, int n, long seed) {
+        return doApproximate(fx, a, b, n, new Random(seed));
+    }
+
+    /**
+     * Approximates the definite integral of a given function over a specified
+     * interval using the Monte Carlo method with a random seed based on the
+     * current system time for more randomness.
+     *
+     * @param fx    the function to integrate
+     * @param a     the lower bound of the interval
+     * @param b     the upper bound of the interval
+     * @param n     the number of random samples to use
+     * @return      the approximate value of the integral
+     */
+    public static double approximate(Function<Double, Double> fx, double a, double b, int n) {
+        return doApproximate(fx, a, b, n, new Random(System.currentTimeMillis()));
+    }
+
+    private static double doApproximate(Function<Double, Double> fx, double a, double b, int n, Random generator) {
+        if (!validate(fx, a, b, n)) {
+            throw new IllegalArgumentException("Invalid input parameters");
+        }
+        double totalArea = 0.0;
+        double interval = b - a;
+        for (int i = 0; i < n; i++) {
+            double x = a + generator.nextDouble() * interval;
+            totalArea += fx.apply(x);
+        }
+        return interval * totalArea / n;
+    }
+
+    private static boolean validate(Function<Double, Double> fx, double a, double b, int n) {
+        boolean isFunctionValid = fx != null;
+        boolean isIntervalValid = a < b;
+        boolean isSampleSizeValid = n > 0;
+        return isFunctionValid && isIntervalValid && isSampleSizeValid;
+    }
+}