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Math Help - deriving length of segment

  1. #1
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    deriving length of segment

    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?

    thanks!
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  2. #2
    MHF Contributor

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    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 " x_i" then the other endpoint is " x_{i+1}". The corresponding points on the curve are (x_i, f(x_i)) and (x_{i+1}, f(x_{i+1}) and the straight line distance between them is \sqrt{(x_{i+1}- x{i})^2+ (f(x_{i+1})- f(x_i))^2}. The total length of all such segments, and so an approximation to the arclength is \sum_{i=0}^{n-1} \sqrt{(x_{i+1}- x_i)^2+ (f(x_{i+1}- f(x_i))^2}.

    We can simplify that a little by factoring a " (x_{i+1}- x_i)^2" out of the square root- of course, it is no longer squared outside the square root: \sum_{i=0}^{n-1}\sqrt{1+ \frac{(f(x_{i+1}- f(x_i))^2}{(x_{i+1}- x_i)^2}}(x_{i+1}- x_i).

    Now let \Delta x_i= x_{i+1}- x_i and \Delta y_i= f(x_{i+1})- f(x_i) and that sum becomes \sum_{i=0}^{i-1}\sqrt{1+ \left(\frac{\Delta y_i}{\Delta x_y}\right)^2}\Delta x_i.

    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
    \int_{x_1}^{x_2} \sqrt{1+ \left(\frac{dy}{dx}\right)^2} dx
    because the fraction \frac{f(x_{i+1})- f(x_i)}{x_{i+1}- x_{i}}= \frac{\Delta f(x_i)}{\Delta x_i}, a "difference quotient", becomes the derivative.
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  3. #3
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    for the summation, in your above equation, why is it (i-1) and not (n-1)?

    what happens to the change in xi as you take summation?

    thanks thanks!
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