Remember how this algorithm works. We approximate the area under the curve with a trapezoid. Then we decide whether or not to accept this approximation by calculating the areas of the two smaller trapezoids formed using the midpoint. If their sum is close enough to the original trapezoid, we accept the original trapezoid as an approximation to the area under the curve. If the sum is not close enough, then we take each of the two smaller trapezoids and apply the above algorithm recursively to each. Start a new thread each time the algorithm is called recursively. See the skeleton code below.

The input data consists of three `double` numbers:
*a*, *b*, and *epsilon*.
The function to integrate is defined in a separate class.
You may parse these three numbers from the command line
or read them from the keyboard or a file.

Try your program on these functions (*epsilon* of 0.0001):
*sin(x)* from 0.1 to 1.0,
*tan(x)* from 0.1 to 1.0,
*x**2* from 0.1 to 1.0,
*sqrt(x)* from 0.1 to 1.0,
and
*-log(x)* from 0.1 to 1.0 (note this is the negative of the logrithm).
**Include these examples** in your submitted sample output.

This algorithm coded in Java might not work well from 0.0 to *b*
on some functions whose value at 0.0 is 0.0,
so that is why 0.1 is used above.
Since this is just a warm-up exercise and not a numerical methods class,
we will not worry about this.

interface TheFunction { public double evaluate(double x); } class MyFunction implements TheFunction { public double evaluate(double x) { return x*x; } } class Sine implements TheFunction { public double evaluate(double x) { return Math.sin(x); } } class Tangent implements TheFunction { public double evaluate(double x) { return Math.tan(x); } } class AdaptiveQuad { public static void main(String[] args) { double a = (Double.valueOf(args[0])).doubleValue(); double b = (Double.valueOf(args[1])).doubleValue(); double epsilon = (Double.valueOf(args[2])).doubleValue(); TheFunction fn; double result; ... fn = (TheFunction) new MyFunction(); Area area = new Area(a, b, epsilon, fn); area.start(); try { area.join(); } catch (InterruptedException e) {} result = area.getResult(); ... fn = (TheFunction) new Sine(); ... fn = (TheFunction) new Tangent(); ... } } class Area extends Thread { private double p, q, epsilon, result; private TheFunction f; public Area(double a, double b, double eps, TheFunction fn) { ... } public double getResult() { return result; } public void run() { if ... // the area in trapezoid [p,q] is close to the sum of the ... // areas in trapezoids [p,(p+q)/2] and [(p+q)/2,q]; ... // use relative error to determine how close, that is, ... // x and y are close if abs((y-x)/x) < epsilon ... // to avoid dividing by zero, use abs(y-x) <= epsilon*x result = ... // the area of trapezoid [p,q] else { ... new Area(p, (p+q)/2, epsilon, f); ... // start a new thread ... new Area((p+q)/2, q, epsilon, f); ... // start a new thread result = ... // call getResult() twice } } }

Animate your program using
`XtangoAnimator`.