In three dimensions, write x, y, z as functions of parameter t and do the same:
[tex]\int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2+ \left(\frac{dy}{dt}\right)^2+ \left(\frac{dz}{dt}\right)^2}dt[/tex]
By using exactly the same formulas but converting from the other coordinate system.
For example, suppose a path is given in spherical coordinates by [itex]\rho= 1[/itex], [itex]\phi= \pi/3[/itex], [itex]\theta= t[/itex], with parameter t. In Cartesian coordinates that is [itex]x= \rho cos(\theta) sin(\phi)= (\sqrt{3}/2)cos(t)[/itex], [itex]y= \rho sin(\theta)sin(\phi)= (\sqrt{3}/2)sin(t)[/itex], [itex]z= \rho cos(t)= cos(t)[/itex].
And my only criticism is that you should stop apologizing for your English. It is excellent. Far better than my (put whatever language you wish in here!).