%% Approximates the Arc Length of the curve y = sqrt(16-4x^2} 
%% on the interval [0,1]. 
clc
format compact 
format short
a=0, b=1, n=10, dx1=(b-a)/n
x1=linspace(a,b,n+1) % Break up the interval into n=10 parts 
y1=2*sqrt(4-x1.^2) % Determines the height at each point x1 
dist1 = sqrt(diff(x1).^2+diff(y1).^2)
% Computes distances between points
format long g
arclength1  = sum(dist1) % sums the distances 
% Alternate method using ds=sqrt(1+(dy/dx)^2)dx, 
% Where dy/dx = (-2x)/(sqrt(4-x^2)), and ds = sqrt((4+3x^2)/(4-x^2)) 
format short 
ds = sqrt((4+3*x1.^2)./(4-x1.^2)) 
format long g
otherarclength1 = dx1*(sum(ds)-(ds(1)+ds(end))/2) 
abs(arclength1-otherarclength1) 
exact = 1.1706509330085367745
% The exact answer (to 9 decimals) using elliptic integrals.