Equivalent section charactistics are generally designed for bending stress and then checked for shear and deflection. Cutting does not decrease strength (ignoring size factors) it decreases section modulus.
Using allowable (permissible) stress design, the required elastic section modulus, Z, is found by dividing the maximum bending moment by the adjusted allowable bending stress, Fb. As we all should know Z is a function of height squared and width. You can not use an area fuction, but must use Fb * Z equivalancy.
Live load deflection = 5/384 [wL / (EI)] gives us the functional inverse realtionship E * I