The AdS/CFT correspondence states that certain 3-dimensional conformal field theories at strong coupling are dual to 4-dimensional AdS space times a compact 7-dimensional Einstein space Y. I will explain how one can test these dualities by calculating some large-N matrix integrals that were derived on the field theory side of the correspondence and from which one can extract the volumes of the various spaces Y. Most of my talk will be based on arXiv:1011.5487, so I will be discussing mostly theories with at least N=3 supersymmetry. I will also mention possible generalizations to theories with N=2 supersymmetry.