can someone teach me how to derive the integration of finding the length of a segment from (x1, y1) to (x2, y2) by spliting the x-axis into steps and using limits and summations?
It would be better to ask about a specific example than rather than just the general concept! Also, I take it you are actually asking about the arclength of a curve rather than just the length of a straight line segment since that would not require "limits and summations".
Suppose y= f(x) gives a graph that passes through (x1,y1) and (x2,y2). Divide the x-axis into n smaller segments (not necessarily of the same length but that would simplify calculations if you were actually doing calculations).
If one endpoint of such an interal is " " then the other endpoint is " ". The corresponding points on the curve are and and the straight line distance between them is . The total length of all such segments, and so an approximation to the arclength is .
We can simplify that a little by factoring a " " out of the square root- of course, it is no longer squared outside the square root: .
Now let and and that sum becomes .
Of course, that is a "Riemann sum" and in the limit, n goes to 0 so we are taking more and more small segments, it becomes the integral
because the fraction , a "difference quotient", becomes the derivative.