if $U$ and $V$ are two open subset of $\mathbb{R}^{n}$, $\varphi:U\rightarrow V$ a $C^{1}$ diffeomorphism, then we have the change of variable formula for the Lebesgue integral: $$\int_{V}f\mathsf d\lambda=\int_{U}(f\circ \phi)\times |J_{\varphi}|\mathsf d\lambda .$$
Now, if I want to compute the Lebesgue integral on a parametrized curve $C$, with $\varphi:I\subset\mathbb R \rightarrow C\subset\mathbb R^{n}$, how do I obtain the following formula: $$\int_{C}f(y)\mathsf dy=\int_{I}f(\varphi(x)) \|\triangledown \varphi(x)\|\mathsf dx$$