Let $C$ denote a curve in $\mathbb{R}^2$ with parameterization $x_C(t), y_C(t), t \in [0,1]$

A common error is transforming the line integral $$ \int_C f(x,y) dC \rightarrow \int_0^{1} f(x_C(t), y_C(t)) dt $$

Which is wrong since it doesn't take into account the rescaling of distance when parametrizing over the curve.

But that leads to an interesting question, what exactly does this quantity measure? Its easy to argue what the definition of a line integral should be by making an argument alone the lines of $(dC)^2 = (dx)^2 + (dy)^2$, but if you wanted to start with that incorrect integral above, and show why it isn't the "curtain" one would expect what's the best way to proceed?


I think it is easy to see that there is no obvious meaning to this, if you consider the one-variable analog. That would be to have an interval $[a,b]$, a function $g:[0,1]\to[a,b]$, and then compare $$ \int_a^bf(x)\,dx $$ with $$ \int_0^1 f(g(t))\,dt. $$


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