In this paper, we segment the volume into geometrically disjoint regions that can be rendered to provide a more effective and interactive volume rendering of structured and unstructured grids. Our segmentation is based upon intervals within the scalar field, producing a set of geometrically defined interval volumes. We present many advantageous properties in using interval volumes, and provide several new rendering operations or shaders to provide effective visualizations of the 3D scalar field. In particular, we demonstrate new technologies that allow interval volumes to be rendered interactively primitives in a volume renderer. We illustrate the use of interval volumes to highlight contour boundaries or material interfaces. Several surface shaders that can easily be integrated in the volume renderer are presented. To construct the interval volumes, we cast the problem one dimension higher, using a higher-dimensional isosurface construction for interactive computation or segmentation. The algorithm is independent of the dimension and topology of the polyhedral cells comprising the grid, and thus offers an excellent enhancement to the volume rendering of unstructured grids. We present examples using hexahedral and tetrahedral cells from time-varying and multi-attribute datasets.