// an abstract class to represent a one-variable mathematical function // TODO:

Question

 // an abstract class to represent a one-variable mathematical function // TODO: complete the methods integrate() and drawIntegral() below    public abstract class Function {   /*    * TODO    * computes the definite integral from a to b using a rectangle approximation    * the argument n is the number of rectangles to be created    * returns the sum of the areas of the rectangles (the approximate integral)    * the method should use a left-endpoint approximation (the first left endpoint will be at x=a)    */   public double integrate(double a, double b, double n) {     //TODO     }      /*    * TODO    * draws the rectangle approximation of the definite integral from a to b    * the argument n is the number of rectangles to be created    * each rectangle has a base on the x-axis with an equal width    * each rectangle has a height given by f(x)    */   public void drawIntegral(double a, double b, double n) {     //TODO         }            /*******CODE BELOW IS COMPLETE AND SHOULD NOT BE MODIFIED************      /*    * a method to compute f(x), the y-value of this function    * note: this method is abstract and must be implemented by sub-classes    */   public abstract double f(double x);         /*    * a method to print a String representation of this function    * note: this method is abstract and must be implemented by sub-classes    */   public abstract String toString();         /*    * A method to draw a graph by connecting the points given    * by the x-values and y-values arrays    */   public void drawGraph(double x[], double y[]) {     //"connect the dots" by drawing lines between successive points in the given arrays     for (int i=0; i<x.length-1; i++) {       StdDraw.line(x[i],y[i], x[i+1], y[i+1]);     }   }      /*    * this method approximates f'(x) using the limit definition of the derivative    * the argument h should be a small value to get a decent approximation    */   public double fprime(double x, double h) {     return (f(x+h)-f(x))/h;    }      /*    * this method fills the given arrays with x-values and y-values    * corresponding to points on the graph of f(x)    * x-values start at xMin and increase by the given step size    * until the array is full.    */   public void computePoints(double x[], double y[], double xStart, double step) {     //first point     x[0] = xStart;     y[0] = f(xStart);          //remaining points     for (int i=1; i<=x.length-1; i++) {       x[i] = x[i-1] + step;       y[i] = f(x[i]);     }   } } 

Explanation / Answer

// an abstract class to represent a one-variable mathematical function
// TODO: complete the methods integrate() and drawIntegral() below

public abstract class Function {
/*
* TODO
* computes the definite integral from a to b using a rectangle approximation
* the argument n is the number of rectangles to be created
* returns the sum of the areas of the rectangles (the approximate integral)
* the method should use a left-endpoint approximation (the first left endpoint will be at x=a)
*/
public double integrate(double a, double b, double n) {
double h = (b - a) / n; // step size

// 1/3 terms
double sum = 1.0 / 3.0 * (f(a) + f(b));

// 4/3 terms
for (int i = 1; i < n; i += 2) {
double x = a + h * i;
sum += 4.0 / 3.0 * f(x);
}

// 2/3 terms
for (int i = 2; i < n; i += 2) {
double x = a + h * i;
sum += 2.0 / 3.0 * f(x);
}

return sum * h;
}
  
/*
* TODO
* draws the rectangle approximation of the definite integral from a to b
* the argument n is the number of rectangles to be created
* each rectangle has a base on the x-axis with an equal width
* each rectangle has a height given by f(x)
*/
public void drawIntegral(double a, double b, double n) {
//TODO
       double h = (b - a) / n; // step size

// 1/3 terms
double sum = 1.0 / 3.0 * (f(a) + f(b));

// 4/3 terms
for (int i = 1; i < n; i += 2) {
double x = a + h * i;
double y= f(x);
       System.out.println("x,y"+x+","+y);
}

// 2/3 terms
for (int i = 2; i < n; i += 2) {
double x = a + h * i;
double y= f(x);
       System.out.println("x,y"+x+","+y);
}

   
}
  
  
  
/*******CODE BELOW IS COMPLETE AND SHOULD NOT BE MODIFIED************
  
/*
* a method to compute f(x), the y-value of this function
* note: this method is abstract and must be implemented by sub-classes
*/
public abstract double f(double x);
  
  
/*
* a method to print a String representation of this function
* note: this method is abstract and must be implemented by sub-classes
*/
public abstract String toString();
  
  
/*
* A method to draw a graph by connecting the points given
* by the x-values and y-values arrays
*/
public void drawGraph(double x[], double y[]) {
//"connect the dots" by drawing lines between successive points in the given arrays
for (int i=0; i<x.length-1; i++) {
StdDraw.line(x[i],y[i], x[i+1], y[i+1]);
}
}
  
/*
* this method approximates f'(x) using the limit definition of the derivative
* the argument h should be a small value to get a decent approximation
*/
public double fprime(double x, double h) {
return (f(x+h)-f(x))/h;
}
  
/*
* this method fills the given arrays with x-values and y-values
* corresponding to points on the graph of f(x)
* x-values start at xMin and increase by the given step size
* until the array is full.
*/
public void computePoints(double x[], double y[], double xStart, double step) {
//first point
x[0] = xStart;
y[0] = f(xStart);
  
//remaining points
for (int i=1; i<=x.length-1; i++) {
x[i] = x[i-1] + step;
y[i] = f(x[i]);
}
}
}

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