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Arc length

  1. Mar 14, 2015 #1
    The arc length of any curve defined by ##y = f(x)## is found as follows:
    $$ds = \sqrt{dx^2 + dy^2}$$
    $$ds = \sqrt{dx^2(1 + {\frac{dy}{dx}}^2)}$$
    $$ds = \sqrt{dx^2} \sqrt{1 + [f'(x)]^2}$$
    $$ds = \sqrt{1 + [f'(x)]^2} dx$$
    Isn't ##\sqrt{dx^2}## equal to ##|dx|##, and not ##dx##?
     
  2. jcsd
  3. Mar 14, 2015 #2

    mfb

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    In principle it is, but if you go in positive x-direction only it does not matter. That way of dealing with differentials is not very mathematical anyway.
     
  4. Mar 15, 2015 #3
    I'm not very familiar with the notion of "going in the positive direction" while plotting a function, I had no idea it makes a difference. Could you please elaborate?
     
  5. Mar 15, 2015 #4

    mfb

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    "dx>0"

    If you want to know the arc length between x=2 and x=4 for example, you integrate x from 2 to 4 and not from 4 to 2.
     
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