Integrating both sides and solving for Φ e for a material of real thickness ℓ, with the incident radiant flux upon the slice Φ Φ e i = Φ Φ e ( 0 ) and the transmitted radiant flux Φ Φ e t = Φ Φ e ( ℓ ℓ ) gives Φ Φ e t = Φ Φ e i exp ⁡ ⁡ ( − − ∫ ∫ 0 ℓ ℓ μ μ ( z ) d z ) , and finally T = Φ Φ e t Φ Φ e i = exp ⁡ ⁡ ( − − ∫ ∫ 0 ℓ ℓ μ μ ( z ) d z ) .