# Line integral in proof of Green's theorem

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

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).

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.