Ho do I use symbolic expressions to define a new function?

Hello everyone,

i have a problem when trying define a new function and integrate it. Say I have a continously differentiable function f:[a,b] -> R and want to calculate its arc length. I would define define a function e.g.

f = @(t) cos(t)

and then, attempting to use the formula L = integral_a^b(sqrt(1+f'(t)^2)dt i would use

df = diff(f, t)

to determine its (symbolic) differential and then define a new function

g = @(t) sqrt(1+df.^2)

and integrate g

integral(g, a, b)

At least thats what i want to do, because it doesnt work, because something is wrong with/MATLAB doesnt unterstand the way i define g. It does not substitute in the symbolic expression previously derived for df and hence throws errors at me when trying to calculate the integral. The problem would also arise if i simply tried to calculate the integral of the derivate. Hence my question: How do i use diff(f, t) to define a new function (which i can the integrate for example).

Answer 1

[Star Strider]

See if this slight revision of the posted code does what you want:

f = @(t) cos(t);
% L = integral_a^b(sqrt(1+f'(t)^2)dt
df = @(f,t) (f(t+1E-8)-f(t-1E-8))/2E-8;         % Numeric Approximation
g = @(t) sqrt(1+df(f,t).^2);
a = 0;
b = 2*pi;
G = integral(g, a, b)

G = 7.6404

.

[Ronald Koelpin]

That does work, thank you. I also found another way:

f = cos(t);
df = diff(matlabFunction(f, 'vars', t), t);
g = matlabFunction(sqrt(1+df.^2), 'vars', t);
ArcLen = integral(g, 0, 2*pi)

[Star Strider]

As always, my pleasure!

The code you posted uses anonymous functions, not symbolic functions, so I stayed with that. The Symbolic Math Toolbox is certainly appropriate here as well.

.