We study the effect of simultaneous bounds on the local L 1 norms of the second fundamental form and of the Gauss curvature on the geometry of surfaces 7 embedded in a Riemannian manifold M. Such bounds are natural since (together with an area bound) they amount to a local bound on the area of the manifold of unit normals to 7, living in the sphere bundle of M. The main technical point is that, given a sequence of surfaces 7 i with uniform local bounds of this type converging in the Hausdorff metric topology to a surface 7 0 , the length space structures of the 7 i converge to that of 7 0 (i.e., distances do not collapse). It follows that under suitable local topological conditions the surface 7 0 is a manifold of bounded integral curvature (MBC), in the sense of Alexandrov, under all smooth complete changes of the metric on M. In particular the boundary of a convex set with nonempty interior in any smooth complete M 3 is MBC. Our method is to construct local coordinates for smoothly embedded surfaces 7 satisfying W 1, BV bounds based solely on curvature integrals. In the process we demonstrate a regularity property for cut loci of surfaces.