This is given in “Vector Analysis Problem Solver” chapter 16, the chapter on volume integrals. I use example 16.3 of VAPS to justify the correct expression for the intensity of black body radiation from the Planck distribution. This is Eq. (16) and has two terms missed entirely by the old theory. I worked out the integration mathematics with close logic by directly adapting the mathematics of volume integrals. Unless there is an error in this close logic the usual expression for the Stefan Boltzmann law is completely wrong. Rayleigh gave no justification in 1900 for his neglect of higher order infinitesimals, Jeans simply made a proportionality correction in 1905.