Question about arc length and the condition dx/dt > 0

songoku
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That passage is from James Stewart (Multivariable Calculus). I want to ask about the condition dx/dt > 0. If dx / dt < 0, the formula can't be used?

Thanks
 
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Why not ? Can you find a counter-example ?
 
BvU said:
Why not ? Can you find a counter-example ?
In my opinion, it can because from the derivation I don't see the need for dx/dt to be positive.

I just don't understand why dx / dt > 0 is written there in the text.

Thanks
 
songoku said:
I just don't understand why dx / dt > 0 is written there in the text.
If dx/dt > 0 then then x is increasing. Conversely, if dx/dt < 0 then x is decreasing.

Consider the parametric curve ##x = \sin(t), y = 1## for ##t \in [0, \pi]##. Graph this simple "curve" and determine its arc length from your graph. What does the integral formula for arc length of this curve produce?
 
The length of a curve should be the same, whether you start measuring it from the left end (dx/dt &gt; 0) or the right end (dx/dt &lt; 0). Some details of the derivation will change if dx/dt &lt; 0; in particular the assumption that f(\alpha) = a &lt; b = f(\beta) must be replaced by f(\alpha) = b &gt; a = f(\beta).
 
Mark44 said:
If dx/dt > 0 then then x is increasing. Conversely, if dx/dt < 0 then x is decreasing.

Consider the parametric curve ##x = \sin(t), y = 1## for ##t \in [0, \pi]##. Graph this simple "curve" and determine its arc length from your graph. What does the integral formula for arc length of this curve produce?
The graph is horizontal line y = 1 and 0 ≤ x ≤ 1.

I think the arc length should be 2 because the curve is traversed twice, once from left to right for 0 ≤ t ≤ π/2 and then from right to left for π/2 ≤ t ≤ π.

From integration, I get zero

pasmith said:
The length of a curve should be the same, whether you start measuring it from the left end (dx/dt &gt; 0) or the right end (dx/dt &lt; 0). Some details of the derivation will change if dx/dt &lt; 0; in particular the assumption that f(\alpha) = a &lt; b = f(\beta) must be replaced by f(\alpha) = b &gt; a = f(\beta).
I think I understand more clearly now from your post and Mark's example. The condition dx / dt > 0 is to ensure the curve is traversed once, only from left to right. If I want to use integration to find the arc length, I need to divide it into two cases, for 0 ≤ t ≤ π/2 and π/2 ≤ t ≤ π then subtract.

Thank you very much for the help and explanation BvU, Mark44, pasmith
 
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