How to calculate the length of an arbitrary arc. Full article at http://www.firsttimeprogrammer.blogspot.com/2015/03/how-to-calculate-length-of-arbitrary-arc.html

line_integral.m

% Line integral calculation %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Variable definition for symbolic calculations
syms t;

% Parametric curve
x = @(t) t;
y = @(t) sqrt(t);

% Integration limits
a=0;
b=10;

% Let's define ds
ds = sqrt(diff(x,t)^2+diff(y,t)^2);

% Indefinite integral
S = int(ds,t);

% Definite integral
line = int(ds,t,a,b)

% Plot of the data
points = 200;
xtra = 5;

x1 = linspace(0,b+xtra,points);
plot(x1,y(x1),'b.');
grid on;

x2 = linspace(a,b,(b+xtra)/b*points);
hold on
plot(x2,y(x2),'r.');
hold off

title('Line integral')
xlabel('x')
ylabel('y')