Let $G$ be a compact Lie group and let $H$ be a Lie subgroup of $G$ which acts on a finite dimensional vector space $V$.
Let's consider the action of $H$ on $G \times V$: $$h((g,v)):= (gh,h^{-1}v), \quad h \in H , g \in G , v \in V, $$ and define the manifold $M$ to be the quotient $G \times_H V$. Then the projection $M = G \times_H V \rightarrow B= G/H$ is a vector bundle.
If $\alpha $ is a differential form on $M$ and $\int_{M/B} \alpha$ denotes the integration along fibers of the $\alpha$, could you please explain to me how can we integrate $\alpha$ along fibers ?