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In wikipedia page about Green's theorem the following equality appears:

$$ \int_{C_1} L(x,y)\, dx = \int_a^b L(x,g_1(x))\, dx $$

I do not understand it. Wikipedia page about line integral defines line integral, when applied to an scalar function, as:

$$ \int_{\mathcal{C}} f(\mathbf{r})\, ds = \int_a^b f\left(\mathbf{r}(t)\right)|\mathbf{r}'(t)| \, dt. $$

that, applied to the proof expression and taken into account that the curve $C_1$ has been parametrized as $(x,g_1(x))$, gives (?):

$$ \int_{C_1} L(x,y)\, dx = \int_a^b L(x,g_1(x)) \,\, |(1,g_1'(x)| \,\, dx $$

that seems different to the one said in the proof (all curve derivative term has been supresed).

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Note that one integral is a $ds$ integral and the other integral is a $dx$ integral. Here, Green's Theorem is written in the $\int L\,dx+M\,dy$ form.

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