Adaptive numerical quadrature is used to normalize posterior distributions in many Bayesian models. We provide the first stochastic convergence rate for the error incurred when normalizing a posterior distribution under typical regularity conditions. We give approximations to moments, marginal densities, and quantiles, and provide convergence rates for several of these summaries. Low- and high-dimensional applications are presented, the latter using adaptive quadrature as one component of a more sophisticated approximation framework, for which limited theory is given. Extension of the theory to the high-dimensional framework for the Laplace approximation (a specific instance of an adaptive quadrature method) is considered and guarantees are provided under additional regularity assumptions