Michael Corral: Vector Calculus

114 CHAPTER 3. MULTIPLE INTEGRALS
with the sums taken over the N random points (x 1 ,y 1 ),..., (x N ,y N ). The±“error term”
in formula (3.15) does not really provide hard bounds on the approximation. It represents
a single standard deviation from the expected value of the integral. That is, it
provides a likely bound on the error. Due to its use of random points, the Monte Carlo
method is an example of a probabilistic method (as opposed to deterministic methods
such as Newton’s method, which use a specific formula for generating points).
For example, we can use formula (3.15) to approximate the volume V under the
plane z=8x+6y over the rectangle R=[0,1]×[0,2]. In Example 3.1 in Section 3.1,
we showed that the actual volume is 20. Below is a code listing (montecarlo.java) for
a Java program that calculates the volume, using a number of points N that is passed
on the command line as a parameter.
//Program to approximate the double integral of f (x,y)=8x+6y
//over the rectangle [0 ,1]x[0 ,2].
public class montecarlo {
public static void main(String [] args) {
//Get the number N of random points as a command−line parameter
int N = Integer . parseInt(args [0]);
double x = 0; //x−coordinate of a random point
double y = 0; //y−coordinate of a random point
double f = 0.0; //Value of f at a random point
double mf = 0.0; //Mean of the values of f
double mf2 = 0.0; //Mean of the values of f^2
for (int i=0;i