Mining of the enclosed regions that are visited frequently by moving objects (i.e. hot region) is a critical premise for the discovery of movement patterns from trajectory databases, and restricting their coverage is the key to promote precision and efficiency for representation of trajectory patterns. Given a trajectory database, this paper studies how to discover these hot regions and how to constraint their size. A definition of hot region query with coverage constraints is presented with a filter-refinement framework to construct them. In the filter step, the study introduces a grid-based approximate schema to construction the dense regions efficiently; and in the refinement step, the study proposes two trend-based and dissimilarity-based measures, and designs corresponding algorithms and heuristic parameter selection method to rationally reconstruct the regions under the coverage constraints. Experiments on practical datasets validate the effectiveness of this work.