Abstract: Let $G_q\mathbb{R}^{n})$ be the Grassmannian of all linear $q$ dimensional subspaces of $\mathbb{R}^{n}$ and $I$ an integral invariant of $p+q-n$ dimensional submanifolds of $\mathbb{R}^{n}$. Then we give methods of evaluating Crofton type integral $ \int_{G_q(\mathbb{R}^{n})} I(M\cap L)\,{\rm d}L. $ The methods also work for various generalizations of $G_q(\mathbb{R}^{n})$ such as complex Grassmannians.

Key words: Kinematic formulas, Crofton formulas