Let $X$, $Y$ be smooth manifolds and let $\pi \colon X \to Y$ be a submersion. Then for every $y \in Y$ the set $W_y = \pi^{-1}(y)$ is a submanifold in $X$. Let $\mu \in \Gamma(|\Lambda| X)$ be a density on $X$ with compact support. How does one define its pushforward $\pi_* \mu$ then?
If we choose such local coordinates $(x_1,\ldots,x_n)$ on $X$ and $(y_1,\ldots,y_k)$ on $Y$ that $\pi(x_1,\ldots,x_n) = (x_1,\ldots,x_k)$ and $W_y$ is given by $(y_1,\ldots,y_k,x_{k+1},\ldots,x_n)$ (where $y = (y_1,\ldots,y_k)$) then we will have $\mu = \mu(y_1,\ldots,y_k,x_{k+1},\ldots,x_n)$ on $W_y$. If the support of $\mu$ is sufficiently small we can define $$ \widetilde \mu(y_1,\ldots,y_k) = \int\limits_{\mathbb R^{n-k}}\mu(y_1,\ldots,y_k,x_{k+1},\ldots,x_{n})dx_{k+1}\ldots dx_n. $$ Or, if we denote by $i_y \colon W_y \to X$ the inclusion, then $\widetilde \mu(y) = \int_{W_y} i_y^* \mu$. This definition is invariant with respect to the choice of coordinates as above and defines a density on $Y$. In general we can use a partition of unity. Are these considerations correct and is it true that $\pi_\ast \mu = \widetilde \mu$?