Derivation and Percent Error calculation, http://imgur.com/a/r7n0z

derivation.java

import java.text.DecimalFormat;
import java.text.NumberFormat;

// Solution for the question here -> http://imgur.com/a/r7n0z

public class Demo {
	public static void main(String[] args) {
		NumberFormat formatter = new DecimalFormat("#0.00000");
		final double T = 70;
		final double L = 1;
		final double G = 77.2;
		final double c = 0.150;
		final double exact_coef = 0.18568;
		final double[] n_list = new double[] {4, 8, 20, 100};

		System.out.println("n \taproximate \texact \t\tpercent error");
		
		for(int m = 0; m < 4; ++m) {
			final double exact_exp = exact_coef * T * L / (G * Math.pow(c, 4));
			final double n = n_list[m];
			double approx = .0;
			
			// Derivation for n times
			for (int i = 1; i <= n; ++i) {	
				double ri = (n + i - 0.5) * (c / n);
				double Ji = (Math.PI * 0.5) * Math.pow(ri, 4);
				approx += T * (L / n) / (G * Ji);
			}

			final double percent_error = 100 * (approx - exact_exp) / exact_exp; // Save percent error
			
			// Print the result
			System.out.println((int)n_list[m] + " \t" + formatter.format(approx) + " \t" + exact_coef + "\t\t" + formatter.format(percent_error));
		}
	}
}