In this talk, I will explain a technique to compute the expectation of the volume of a random submanifold $Z$ given by the zero set of a nice enough random field. The Kac Rice formula tells us that this is the integral of a density over the ambient manifold $M$. In a joint work with Michele Stecconi, we define a family of convex bodies, the "zonoid section", associated to $Z$, indexed by the ambient manifold $M$ such that the Kac Rice density of $Z$ is given by the "first intrinsic volume" of this convex body. We then show that the zonoid section of the independent intersection $Z\cap Z'$ is determined by the zonoid sections of $Z$ and $Z'$ and that they behave well under pull backs as well (which is not the case for the density alone). Thus these convex bodies contain all the information one needs for expectation of volumes. I will also explain how this zonoid section can be interpreted as the "expected varifold" of the submanifold $Z$ which is a promising lead to generalize these results beyond random fields.