Different results from vpaintegral() and int() on piecewise expression.

syms theta

create_polar_curve = @(theta, params) symfun(...
piecewise(params(2) < theta < pi-params(2),...
    params(1) + params(3) * (-1+cos( (2*pi)/(pi-2*params(2)) * (theta-params(2)))),...
    pi + params(2) < theta < 2*pi-params(2),...
    params(1) + params(3) * (-1+cos( (2*pi)/(pi-2*params(2)) * (theta-params(2)-pi))),...
    params(1) + theta*0 ...
    ), theta);

params = [25 1 2];
r = create_polar_curve(theta, params);

double(arc_length(r,[0 1],[2*pi 4]))
double(arc_length_numerical(r,[0 1],[2*pi 4]))

function length_vector = arc_length(r, theta_1, theta_2)
    syms theta;
    integral(theta) = int(sqrt(r(theta)^2 + diff(r(theta))^2 ),thet
    length_vector = integral(theta_2)-integral(theta_1);
end

function length_vector = arc_length_numerical(r, theta_1, theta_2)
    syms theta;
    length_vector = zeros(1,numel(theta_1));
    for i = 1:numel(length_vector)
        length_vector(i) = vpaintegral(sqrt(r(theta)^2 + diff(r(theta))^2 ),theta,theta_1(i),theta_2(i));
    end
end

output :

ans =

  157.0796   75.0000


ans =

  155.4071   74.1637

>> 

The script above creates a symbolic function representing a curve in polar coordinates, and calculates the arc length of that curve over two intervals. For some reason, computing the integral symbolically is giving slightly incorrect results. I would much prefer to use the symbolic integral, as I need to compute with thousands of different limits of integration. Computing the integral once, and evaluating it multiple times is far faster than approximating the integral multiple times.

Is there any reason int() shouldn't be working here?

[Karan Nandankar]

Hi,

For complex expressions, int() evaluates numerical approximation for the definite integral of the function. You can refer to the documentation below for more information:

https://in.mathworks.com/help/symbolic/int.html#btydjvs-2