Before rendering a translucent volume, we need to understand how such a volume transports light. To this end, we will build an optical model. The optical model describes the light transport within the volume and allows us to define the behavior of light passing through the volume.
Many researchers began building optical models in the early 1980's to synthesize photorealistic images with volumetric elements such as clouds [4,40,62]. By 1988, others were building models to perform volume rendering for scientific visualization [20,83]. Williams and Max  later refined the approach for use with various interpolation functions and cell shapes.
In this chapter, I discuss the volume rendering integral. The volume rendering integral is an equation that computes the color of light that passes through a volume. I first derive the volume rendering integral using a model and derivation similar to that of Max . I then discuss properties of this equation that are important for practical applications. Finally, I present several closed forms that other researchers have developed.