We lay out novel foundations for computer-aided verification of guaranteed bounds on expected outcomes of imperative probabilistic programs featuring (i) general \emph{loops}, (ii) \emph{continuous} distributions, and (iii) \emph{conditioning}. To handle loops we rely on user-provided quantitative \emph{invariants}, as is well established. However, in the realm of continuous distributions, \emph{invariant verification} becomes extremely challenging due to the presence of \emph{integrals} in expectation-based program semantics. Our key idea is to soundly \emph{under} or \emph{over-approximate} these integrals via \emph{Riemann sums}. We show that this approach enables the SMT-based invariant verification for programs with fairly general control flow structure. On the theoretical side, we prove \emph{convergence} of our Riemann approximations, and establish $\mathsf{coRE}$-completeness of the central verification problems. On the practical side, we show that our approach enables to use existing automated verifiers targeting \emph{discrete} probabilistic programs for the verification of programs involving \emph{continuous sampling}. Towards this end, we implement our approach in the recent quantitative verification infrastructure Caesar by encoding Riemann sums in its intermediate verification language. We present several promising case studies.