Let $M \subset \mathbb {R}^d$ be a $n$-dimensional submanifold, $W$ is a coordinate chart of $M$ with the map $\varphi : U \to V$ and $f \colon U \to \mathbb {R}$ is a continuous function function.
Can the Lebesgue integral $\int _U f(x)\,dx$ in $\mathbb {R}^d$ exist if $\int _W f|_W \,dS$ doesn't exist?