Question

Programming Assignment - Numerical Integration In MATH 1775 last semester, you used Riemann sums in Excel to approximate the value of the definite integral that represents the area of the region between the x-axis and the graph of 2 5 y xx = − 8 shown below. The purpose of this assignment is to increase the accuracy of such approximations of integrals. Write a program (using java) that uses the Midpoint Rule, the Trapezoid Rule, and Simpson’s Rule to find separate approximations to a given definite integral ( ) b a f x dx ∫ . The program should accept the following as user inputs: • the limits of integration a and b; • the number of subintervals n, which may be as large as one million; • the coefficients of an arbitrary 5th-degree polynomial 543 2 f x Ax Bx Cx Dx Ex F ( ) = + + + ++ ; Program outputs should include: • the exact value of the integral, as found using the Fundamental Theorem of Calculus. • values of the approximations for each of the three approximation methods; • the percent error for each method.

Answer 1

import java.util.Scanner;

public class Integrate

{

public static Scanner sc = new Scanner (System.in);

public static void main (String[] args) {

System.out.print("Enter the limits of integration a and b : ");

double a = sc.nextDouble();

double b= sc.nextDouble();

  

System.out.print("\nEnter the number of sub-intervasl : ");

int n = sc.nextInt();

  

System.out.print("\nEnter the coefficients of an arbitary 5th degree polynomial: ");

System.out.println("Ax^5 + Bx^4 + Cx^3 + Dx^2 + Ex^1 + F");

System.out.print("\n A = ");

double A = sc.nextDouble();

System.out.print("\n B = ");

double B = sc.nextDouble();

System.out.print("\n C = ");

double C = sc.nextDouble();

System.out.print("\n D = ");

double D = sc.nextDouble();

System.out.print("\n E = ");

double E = sc.nextDouble();

System.out.print("\n F = ");

double F = sc.nextDouble();

  

System.out.print("\nEnter the actual integral value : ");

double actual = sc.nextDouble();

  

System.out.println("Integral value using Mid-point formula : ");

double approx = MidPoint(a, b, n, A, B, C, D, E, F);

System.out.println("Approximate value = " + approx);

System.out.println("Error = " + Error(actual, approx));

  

System.out.println("Integral value using Trapezoidal formula = ");

approx = Trapezoid(a, b, n, A, B, C, D, E, F);

System.out.println("Approximate value = " + approx);

System.out.println("Error = " + Error(actual, approx));

  

  

System.out.println("Integral value using Simpson formula = ");

approx = Simpson(a, b, n, A, B, C, D, E, F);

System.out.println("Approximate value = " + approx);

System.out.println("Error = " + Error(actual, approx));

}

  

public static double f(double x, double A, double B, double C, double D, double E, double F) {

double x2 = x*x;

double x3 = x*x*x;

double y = A*x3*x2 + B*x2*x2 + C*x3 + D*x2 + E*x + F;

return y;

}

public static double MidPoint (double a, double b, int n, double A, double B, double C, double D,

double E, double F) {

double h = (b - a)/n;

double x, x0 = a, x1;

double sum = 0;

  

for (int i = 1; i <= n; i++){

x1 = x0 + h;

x =(x0 + x1)/2;

sum += f(x, A, B, C, D, E, F);

x0 = x1;

}

  

return sum * h;

}

public static double Trapezoid (double a, double b, int n, double A, double B, double C, double D,

double E, double F) {

double h = (b - a)/n;

double x = a;

double sum = 0;

for (int i = 1; i < n; i++)

{

x = a + i * h;

sum += f(x, A, B, C, D, E, F);

}

sum = 2 * sum + (f(a, A, B, C, D, E, F) + f(b, A, B, C, D, E, F));

return sum * h / 2;

}

  

public static double Simpson (double a, double b, int n, double A, double B, double C, double D,

double E, double F) {

double h = (b - a)/n;

double x = a;

double mid = x + h/2;

double sum = f(x, A, B, C, D, E, F);

int i = 0;

while (i < n - 1)

{

i++;

x = a + i * h;

sum = sum + 4 * f(mid, A, B, C, D, E, F) + 2 * f(x, A, B, C, D, E, F);

mid = x + h/2;

}

sum = sum + 4 * f(mid, A, B, C, D, E, F) + f(b, A, B, C, D, E, F);

return sum * h / 6;

}

public static double Error (double actual, double approx) {

return (Math.abs(actual - approx));

}

}