I'm wondering if you could somehow calculate the length of the curve even if it is not smooth at a given interval.
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Welcome to math.se! You may want to specify what kind of curves you are considering. For example, if curve means continuous map $[0,1]\rightarrow \Bbb R^n$, you may for example consider piecewise linear curves, whose length can be specified by adding up the lengths of the line segments. Those are not smooth in general... – Jonas Linssen May 18 '21 at 12:19
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If my curve makes a sharp turn(x'(t) and y'(t) is 0) at one of the endpoints, say "a"(from [a,b]), does that method still apply? – ameagari May 19 '21 at 02:44
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What I was trying to say is: There are many different definitions of curves, most of them coming with their own definition of length. The curve-integral of cause only applies to smooth/$C^1$-curves. So unless you give more context by editing your post to specify what type of curves you are considering (a smooth curve cannot have non-smooth points) I am afraid your question won’t get a satisfying answer. Besides, how can a curve make a turn in an endpoint? – Jonas Linssen May 19 '21 at 06:31
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I purposefully didn't give context because I do not want to violate academic rules our school enforces. However, you did answer my question accidentally! Your point does make sense: "how can a curve make a sharp turn at an endpoint?" Thank you very much! – ameagari May 20 '21 at 09:09