Abstract: I will describe the Verlinde-Segre relationship for the moduli space of semistable vector bundles over a smooth projective curve C, using a suitable Quot scheme on C. In the coprime rank and degree case, I will also explain that any tautological integral on the moduli space of stable bundles on C can be expressed, using the geometry of Quot schemes on C, as a sum of integrals on r-products of Sym^d(C) for suitable degrees, therefore reducing the calculation to rank one. This is the first talk in a lecture series on recent and ongoing work which explores how to relate the intersection theory of moduli spaces of stables sheaves to that of suitable Quot schemes, leading to an effective way of calculating interesting intersection-theoretic invariants of the moduli space.