We present a study of the influence of different approximation schemes on the convergence rate of volume rendering integral (VRI) numerical approximations. We experimentally evaluate the impact of numerical integration techniques on the rate of convergence to the correct solution of the VRI on a single ray. We report that the discretization of both the inner and outer integrals have influence on the overall convergence rate. Then, we present results related to the (traditional) pre-integrated and second-order pre-integrated algorithms. In practice, we observed that pre-integrated lookup tables provide second and third order convergence rates for the VRI approximation, respectively. Our results also suggest that the convergence rate drops one order of magnitude for the second-order algorithm when lookup tables are numerically computed using low sample rates. Also, the convergence of both algorithms drops to linear when the attenuation within ray segment is neglected.