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How to solve a 2D integral?

I am trying to apply the trapezoid rule for a double integral. I tried many approaches, but I can not get it to work correctly.

static double f(double x) {
    return Math.exp(- x * x / 2);
}

// trapezoid rule
static double trapezoid(double a, double b, int N) {
    double h = (b - a) / N;
    double sum = 0.5 *  h * (f(a) + f(b));
    for (int k = 1; k < N; k++)
        sum = sum + h * f(a + h*k);
    return sum;
}

I understand the method for one variable integral, but I do not know how to do this for a 2D integral, say: x + (y * y). Can someone explain this briefly?

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3 answers

If you intend to use the trapezoid rule, you will do this:

// The example function you provided.
public double f(double x, double y) {
    return x + y * y;
}

/**
 * Finds the volume under the surface described by the function f(x, y) for a <= x <= b, c <= y <= d.
 * Using xSegs number of segments across the x axis and ySegs number of segments across the y axis. 
 * @param a The lower bound of x.
 * @param b The upper bound of x.
 * @param c The lower bound of y.
 * @param d The upper bound of y.
 * @param xSegs The number of segments in the x axis.
 * @param ySegs The number of segments in the y axis.
 * @return The volume under f(x, y).
 */
public double trapezoidRule(double a, double b, double c, double d, int xSegs, int ySegs) {
    double xSegSize = (b - a) / xSegs; // length of an x segment.
    double ySegSize = (d - c) / ySegs; // length of a y segment.
    double volume = 0; // volume under the surface.

    for (int i = 0; i < xSegs; i++) {
        for (int j = 0; j < ySegs; j++) {
            double height = f(a + (xSegSize * i), c + (ySegSize * j));
            height += f(a + (xSegSize * (i + 1)), c + (ySegSize * j));
            height += f(a + (xSegSize * (i + 1)), c + (ySegSize * (j + 1)));
            height += f(a + (xSegSize * i), c + (ySegSize * (j + 1)));
            height /= 4;

            // height is the average value of the corners of the current segment.
            // We can use the average value since a box of this height has the same volume as the original segment shape.

            // Add the volume of the box to the volume.
            volume += xSegSize * ySegSize * height;
        }
    }

    return volume;
}

Hope this helps. Feel free to ask any questions that my code may have (warning: code not verified).

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Many ways to do this.

If you already know this for 1d, you can do it like this:

  • g (x), 1d- f (x, y) x
  • 1d- g (x)
  • :)

, , . . monte carlo.

0

Consider using a class jhplot.F2Dfrom the DataMelt Java program . You can integrate and visualize 2D functions by doing something like:

f1=F2D("x*y",-1,1,-1,1) # define in a range
print f1.integral()
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