If the function was a straight line, you could use the Pythagorean theorem to find it.
sqrt( (y2-y1)^2 + (x2-x1)^2 )
If you assume the arc length was a straight line, and slowly bring the points on the function closer together, eventually you get the change in y approaching the slope of the line times the difference in x's.
The arc length for that infinitesimally small section is then sqrt( ( f'(x)dx )^2 + dx^2 ) = sqrt( dx^2 * (f'(x)^2 + 1 ) = sqrt( f'(x)^2 + 1 ) dx. Where f'(x) is the derivative of f(x).
Integrate that between two points to get the arc length between t